- How to Square A Number
- Perfect Squares
- Negative Numbers
- Negatives Roots
- Perfect Square Roots
- The Square Root Symbol
- Square Root Day
- Learn More
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- Properties of perfect square
- How to Square A Number
- Example: What is 3 squared?
- Squares From to
- Negative Numbers
- Square Roots
- Decimal Numbers
- Negatives
- Example: (−3) squared
- Example: What are the square roots of 25?
- The Square Root Symbol
- Example: What is 25?
- Example: What is √36 ?
- Perfect Squares
- Calculating Square Roots
- Example: what is √10?
- The Easiest Way to Calculate a Square Root
- A Fun Way to Calculate a Square Root
- How to Guess
- Square Root Day
- Squares
- Example: What is 3 squared?
- Square Root
- Definition
- The Square Root Symbol
- Example: What is √36 ?
- Negative Numbers
- Example: What is minus 5 squared?
- Two Square Roots
- Example: Solve w2 = a
- Principal Square Root
- Example:
- In a Nutshell
- Why Is This Important?
- Example: Solve x2 − 9 = 0
- Example: Solve for x in (x − 3)2 = 16
- Square Root of xy
- Example: What is √(100×4) ?
- Example: What is √8√2 ?
- Example: What is √(−8 × −2) ?
- Why does √ = √√ ?
- An Exponent of a Half
- Exercises
- Exercises
- Read the following words paying attention to the pronunciation:
- What is the meaning of √ in maths?
- Where does square root come from?
- How do you calculate square root?
- What is a square root of 144?
- What is the square root in simple terms?
- Who invented math?
- What you mean by root?
- What is the answer for a square root called?
- Does a square root have two answers?
- IS 400 a perfect square?
- Is 27 a cube number?
- Does 0 have a square root?
- Is 1 is a perfect square?
- What is the square root of 64?
- What will be the unit digit of the square of 144?
- Is 144 a cube number?
- Frequently Asked Question:
- Why is the square root of not a negative number?
- Why can’t the square root of a number be negative?
- Why is square root only positive?
- Can you have a negative number in a square root?
- Is a square root positive or negative?
- What is the square of negative 4?
- What is a negative 3?
- Is the square root of negative 3 a real number?
- What is the square of 3?
- Can you find the square root of a negative number?
- What is the square of a negative number?
- What is the square root of a negative 1?
- What is a negative square root called?
- How is the square root of negative 1?
- Can I square root a negative number?
- What is the square root of I where I √ 1?
- Who invented the square root of negative one?
- Square Root
- What is Square Root?
- Square Root Definition
- How to Find Square Root?
- Repeated Subtraction Method of Square Root
- Square Root by Prime Factorization Method
- Finding Square Root by Estimation Method
- Calculating Square Root by Long Division Method
- Square Root Table
- Square Root Formula
- Simplifying Square Root
- Square Root of a Negative Number
- Square of a Number
- How to Find the Square of a Number?
- Squares and Square Roots
- Square Root of Numbers
- FAQs on Square Root
- What is Square Root in Math?
- How to Calculate the Square Root of a Number?
- Can Square Root be Negative?
- How to Find the Square Root of a Decimal Number?
- What is the Square Root Symbol?
- How to Multiply Two Square Root Values Together?
- What is the Formula for Calculating the Square Root of a Number?
- What is the Square and Square Root of a Number?
- Which Method is Used to Find the Square Root of Non-Perfect Square Numbers?
- How to Find a Square Root on a Calculator?
- What are the Applications of the Square Root Formula?
- What does the Square of a Number mean?
- How to Calculate the Square Root of a Negative Number?
- Why is the Square of a Negative Number Positive?
- Square Root of 1
- What Is the Square Root of 1?
- Is Square Root of 1 Rational or Irrational?
- How to Find the Square Root of 1?
- Square Root of 1 Solved Examples
- FAQs on the Square Root of 1
- What is the Value of the Square Root of 1?
- If the Square Root of 1 is 1. Find the Value of the Square Root of 0.01.
- What is the Square Root of -1?
- Square Root
- Prime Factorization Method
- Long Division Method
- Table for Square and Square root of first ten natural numbers
- Square Root Formula
- Sample Problems
- What is a Square Root?
- Why do we use plus or minus in square root?
- Sample Problems

## How to Square A Number

To square a number:

**Multiply the number by itself.**

What is 4 squared?

4 squared = 4 × 4 = 16

“Squared” is often written as a little 2 like this:

This says “*4 Squared equals 16*“

(the little 2 says the number appears twice in multiplying)

## Perfect Squares

The Perfect Squares (also called “*Square Numbers*“) are the squares of the integers:

Perfect Squares From 0^{2} to 10^{2}

## Negative Numbers

We can also square negative numbers.

What happens when we square (−7) ?

(−7) × (−7) = 49

(because a negative times a negative gives a positive)

When we square a negative number we get a positive result.

A square root goes the other way:

**What can we multiply by itself to get this?**

Find a square root of 25

What can we multiply by itself to get 25?

A square root of 25 is 5, because when 5 is multiplied by itself we get 25.

a square root of a number is a value that can be multiplied by itself to give the original number.

## Negatives Roots

We discovered earlier that we can square negative numbers:

(−6) × (−6) = 36

And of course 6 × 6 = 36 too.

So the square root of 36 could be −6 or +6

## Perfect Square Roots

The Perfect roots are the roots of the integers:

## The Square Root Symbol

This is the special symbol that means “*square root*“.

Use it like this:

√49 is 7

we say “square root of 49 equals 7”

What is √81 ?

81 = 9 × 9

in other words when we multiply 9 by itself (9 × 9) we get 81

So the answer is:

√81 = 9

## Square Root Day

The April 4^{th} 2016 is a Square Root Day, because the date looks like 4/4/16

The next after that is the May 5^{th} 2025 (5/5/25)

## Learn More

Simplifying Square Roots

Cube Numbers and Cube Roots

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** **A natural number m is said to be square number or perfect square if it can be expressed In terms of n^{2}, where n is also a natural number. It is necessary to know about square and square root to solve number system related problems.

**Example: **4 is a square number as 4 = 2^{2}

## Properties of perfect square

- All perfect square numbers end with 0, 1, 4, 5, 6 or 9 at unit’s place.
- None of these end with 2, 3, 7 or 8 at unit’s place.
- If a number has 1 or 9 in the unit’s place, then it’s square ends in 1.
- when a perfect square number ends in 6, the number whose square it is will have either 4 or 6 in unit’s place.
**There are 2n non-perfect square numbers between n**between 2^{2}and (n + 1)^{2}

Example:^{2 }and 3^{2}the no. of non-perfect square no. are 2×2 =4 as (5, 6, 7, 8)- sum of first n odd natural numbers is n
^{2}also Remember it for square and square root related terms.

**Example:**1+3+5 = 9 = 3^{2} - Any perfect square number can be represented as sum of odd natural numbers starting from 1.
- Also, Any odd perfect square number can be represented as sum of two consecutive natural number.

**Example:**49 = 24+25, 121 = 60 + 61 - For any natural number (n + 1) × (n – 1) = n
^{2 }– 1.

**Example:**6 x 4 = 5^{2 }– 1 = 24 - Square numbers can only have
**even number of 0’s**at the end.

For any natural number a > 1, we have (2a)^{2}+(a^{2} – 1)^{2}=(a^{2 }+ 1)^{2}

**So, 2a, a ^{2 }– 1 and a^{2}+ 1 forms a Pythagorean triplet**.

By this rule, if any one of the members of Pythagorean triplet is given then you can find the rest. this also necessary for getting knowledge of square and square root.

**Example: **1. 3

^{2}+

^{ }4

^{2 }= 5

^{2}so, 3, 4, 5 are Pythagorean triplet.

*If 5 is one of the members of the Pythagorean triplet then find the rest.*

**Solution: **Try 2a = 5 then a will not be an integer so take next

a

^{2}– 1 = 5 then a

^{2}= 6 again not an integer

try a

^{2}+ 1 = 5 then a

^{2}= 4 thus, a = 2

2a = 4

a^{2} – 1 = 3

Thus, Pythagorean triplet are 3, 4, 5.

It is simply the inverse operation of square. Square and square root are in converse of each other.

**There are many ways to find the square root of any number:**

- By just inversion of square.

**Example:**3^{2 }= 9 and √9 = 3 - By repeated subtraction of odd numbers.
9 – 1 = 8, 8 – 3 = 5, 5 – 5 = 0, by 3 times consecutive subtraction we got zero thus √9 = 3.

- By prime factorization.

√36 = 2x2x3x3 = 2^{2}x 3^{2 }thus square root of 36 is 2×3 i.e. 6 - By division method.

Try to solve examples based on square and square root. So that it will be easy for you to solve problems related to it.

First learn about Squares, then Square Roots are easy.

## How to Square A Number

To square a number: **multiply it by itself**.

### Example: What is 3 squared?

«Squared» is often written as a little 2 like this:

This says **«4 Squared equals 16»**

(the little 2 says

the number appears twice in multiplying)

## Squares From to

## Negative Numbers

We can also square **negative numbers**.

That was interesting!

When we square a **negative** number we get a **positive** result.

Just the same as squaring a positive number:

## Square Roots

A **square root** goes the other way:

3 squared is 9, so a **square root of 9 is 3**

It is like asking:

What can we multiply by itself to get this?

Here are some more squares and square roots:

## Decimal Numbers

It also works for decimal numbers.

Using the sliders:

- What is the square root of
**8**? - What is the square root of
**9**? - What is the square root of
**10**? - What is
**1**squared? - What is
**1.1**squared? - What is
**2.6**squared?

## Negatives

We discovered earlier that we can square negative numbers:

### Example: (−3) squared

(−3) × (−3) = **9**

And of course 3 × 3 = **9** also.

So the square root of 9 could be **−3** or **+3**

### Example: What are the square roots of 25?

(−5) × (−5) = 25

5 × 5 = 25

So the square roots of 25 are **−5** and **+5**

## The Square Root Symbol

We use it like this:

and we say **«square root of 9 equals 3»**

### Example: What is 25?

25 = 5 × 5, in other words when we multiply

5 by itself (5 × 5) we get 25

**So the answer is:**

√25 = 5

But wait a minute! Can’t the square root **also be −5**? Because (−5) × (−5) = **25** too.

- Well the
**square root of 25**could be −5 or +5. - But when we use the
**radical symbol**we only give the**positive (or zero) result**.

### Example: What is √36 ?

Answer: 6 × 6 = 36, so **√36 = 6**

## Perfect Squares

The Perfect Squares (also called «Square Numbers») are the squares of the integers:

Try to remember them up to 12.

## Calculating Square Roots

It is easy to work out the square root of a perfect square, but it

is **really hard** to work out other square roots.

### Example: what is √10?

Well, 3 × 3 = 9 and 4 × 4 = 16, so we can guess the answer is between 3 and 4.

- Let’s try 3.5:
*3.5 × 3.5 = 12.25* - Let’s try 3.2:
*3.2 × 3.2 = 10.24* - Let’s try 3.1:
*3.1 × 3.1 = 9.61* - …

Getting closer to 10, but it will take a long time to get a good answer!

At this point, I get out my calculator and it says:

But the digits just go on and on, without any pattern.

So even

the calculator’s answer is **only an approximation !**

*Note: numbers like that are called Irrational Numbers, if you want to know more.*

## The Easiest Way to Calculate a Square Root

And also use your common sense to make sure you have the right answer.

## A Fun Way to Calculate a Square Root

There is a fun method for calculating a square root that gets more and more accurate each time around:

- Our first attempt got us from 4 to
**3.25** - Going again (
*b to e*) gets us:**3.163** - Going again (
*b to e*) gets us:**3.1623**

And so, after 3 times around the answer is 3.1623, which is pretty good, because:

** 3.1623 x 3.1623 = 10.00014**

### How to Guess

In that case we could think «82,163» has 5 digits, so the square root might have 3 digits (100×100=10,000), and the square root of 8 (the first digit) is about 3 (3×3=9), so 300 is a good start.

### Square Root Day

The 4th of April 2016 is a Square Root Day, because the date looks like **4/4/16**

The next after that is the 5th of May 2025 (5/5/25)

309,310,315, 1082, 1083, 2040, 3156, 2041, 2042, 3154

You might like to read our Introduction to Squares and Square Roots first.

## Squares

### Example: What is 3 squared?

«Squared» is often written as a little 2 like this:

This says **«4 Squared equals 16»**

(the little 2 means

the number appears twice in multiplying, so **4×4**=16)

## Square Root

A **square root** goes the other direction:

3 squared is 9, so a **square root of 9 is 3**

It is like asking:

What can I multiply by itself to get this?

## Definition

Here is the definition:

A square root of x is **a number r** whose square is x:

r^{2} = x

r is a square root of x

## The Square Root Symbol

We can use it like this:

we say «square root of 9 equals 3»

### Example: What is √36 ?

Answer: 6 × 6 = 36, so **√36 = 6**

## Negative Numbers

We can also square negative numbers.

### Example: What is **minus 5 squared**?

- square the 5, then do the minus?
- or square (−5)?

It isn’t clear! And we get different answers:

- square the 5, then do the minus: −(5×5) =
- square (−5): (−5)×(−5) =

So let’s make it clear by using «( )».

That was interesting!

When we square a **negative** number we get a **positive** result.

Just the same as when we square a positive number:

Now remember our definition of a square root?

A square root of x is **a number r** whose square is x:

r^{2} = x

r is a square root of x

And we just found that:

(+5)^{2} = 25

(−5)^{2} = 25

So **both +5 and −5** are square roots of 25

## Two Square Roots

There can be a **positive** and **negative** square root!

This is important to remember.

### Example: Solve w^{2} = a

**w = √a** and **w = −√a**

## Principal Square Root

So if there are really two square roots, why do people say √ =

There **are** two square roots, but **the symbol ** means **just the principal square root**.

### Example:

The square roots of 36 are 6 **and** −6

But √ = (not −6)

The Principal Square Root is sometimes called the Positive Square Root (but it can be zero).

## In a Nutshell

When we have:r^{2} = x

r = ±√x

## Why Is This Important?

Why is this «plus or minus» important? Because we don’t want to miss a solution!

### Example: Solve x^{2} − 9 = 0

x^{2} − 9 = 0

Move 9 to right:x^{2} = 9

x = ±√9

x = ±3

The «» tells us to include the «−3» answer also.

### Example: Solve for x in (x − 3)^{2} = 16

(x − 3)^{2} = 16

x − 3 = ±√16

x − 3 =

Add 3 to both sides:x = 3 ± 4

x = 7 or −1

Check: (7−3)^{2} = 4^{2} = 16

Check: (−1−3)^{2} = (−4)^{2} = 16

## Square Root of xy

When two numbers are multiplied **within** a square root, we can split it into a multiplication of two square roots like this:

√ = √√

but only when **x** and **y** are **both greater than or equal to 0**

### Example: What is **√(100×4)** ?

= √(100) × √(4)

= 10 × 2

And **√√ = √** :

### Example: What is **√8√2** ?

### Example: What is **√(−8 × −2)** ?

√(−8 × −2) = √(−8) × √(−2)

We seem to have fallen into some trap here!

We can use Imaginary Numbers,

but that leads to a **wrong** answer of **−4**

The rule only works when **x** and **y** are both greater than or equal to 0

So we can’t use that rule here.

Instead just do it this way:

√(−8 × −2) = √16 = +4

### Why does √ = √√ ?

We can use the fact that squaring a square root gives us the original value back again:

(√)^{2} = a

Assuming **a** is not negative!

We can do that for xy:(√)^{2} = xy

And also to x, and y, separately:(√)^{2} = (√)^{2}(√)^{2}

Use a^{2}b^{2} = (ab)^{2}:(√)^{2} = (√√)^{2}

Remove square from both sides^{}:√ = √√

## An Exponent of a Half

A square root can also be written as a fractional exponent of one-half:

but only for **x** greater than or equal to 0

To

square a number^{1},

you have learned, you must multiply that number by itself. The square

root of a number is just the opposite. When you find the square root

of a number, you are finding what number multiplied by itself gives

you the number you began with^{2}.

The sign for the square root is √. Thus, the square root of 25 is

represented by √25. 25 is a perfect square. That is, a whole number

(5) multiplied by itself will give you 25. Most numbers are not

perfect

squares.

In that case, to get the square root of a number we may either find

it by taking an arithmetic square root or by using a table.

The

process of finding a root is known as evolution; it is the inverse of

involution, because by the aid of this process we try to find that

which is given only when raising a number to a power (viz. the base

of the power), while the data given is just what is sought for^{3}

raising a number to a power (viz. the power itself). Therefore the

accuracy of the root taken may always be checked by raising the

number to the power^{4}.^{ }For

instance, in order to check the equality: ^{3}√125=5,

it is sufficient to cube 5; obtaining the quantity under the radical

sign,

we conclude that the cube root of 125 has been found correctly.

^{1 }to

square

a

number

— чтобы возвести число в квадрат

^{2}^{ }the

number you began with — зд.

исходное

число

^{3 }what

is sought for — искомое

^{4 }by

raising the number to the power — возведением

числа

в степень

## Exercises

inverse,

learn, perfect, order, for, opposite, not, must, number, thus.

to

square, to use, to raise, to multiply, to find, to check, to give, to

begin, to obtain, to get, to take, to be.

*Make*

up sentences of your own using the words and expressions given

below:

to

raise to power, to obtain the quantity, to square the number, to take

an arithmetic square root, to use a table may be checked, conclude.

1.

What operation should be performed to square a number? 2. What is a

perfect square? 3. What do we do to get the square root of a number?

4. What is the process of finding a root called? 5. How do we check

the accuracy of a root?

**V.**

Translate into Russian:

Tables

of squares are used by architects and engineers in working with

squares of number. If you have a table of squares, you can find the

approximate square root of any number. Sometimes it is not easy to

find a square root by

inspection.

If a table of squares is not at hand another method may be used.

Чтобы

возвести в квадрат число, надо умножить

это число на самое себя. Извлечение

квадратного корня — это действие

обратное возведению в квадрат. Чтобы

получить квадратный корень числа, мы

можем пользоваться специальной таблицей.

Правильность извлечения квадратного

корня можно проверить, возведя в квадрат

подкоренное

выражение;

если получится данное число, то корень

найден правильно.

To

square a number^{1},

you have learned, you must multiply that number by itself. The square

root of a number is just the opposite. When you find the square root

of a number, you are finding what number multiplied by itself gives

you the number you began with^{2}.

The sign for the square root is √. Thus, the square root of 25 is

represented by √25. 25 is a perfect square. That is, a whole number

(5) multiplied by itself will give you 25. Most numbers are not

perfect

squares.

In that case, to get the square root of a number we may either find

it by taking an arithmetic square root or by using a table.

The

process of finding a root is known as evolution; it is the inverse of

involution, because by the aid of this process we try to find that

which is given only when raising a number to a power (viz. the base

of the power), while the data given is just what is sought for^{3}

raising a number to a power (viz. the power itself). Therefore the

accuracy of the root taken may always be checked by raising the

number to the power^{4}.^{ }For

instance, in order to check the equality: ^{3}√125=5,

it is sufficient to cube 5; obtaining the quantity under the radical

sign,

we conclude that the cube root of 125 has been found correctly.

^{1 }to

square

a

number

— чтобы возвести число в квадрат

^{2}^{ }the

number you began with — зд.

исходное

число

^{3 }what

is sought for — искомое

^{4 }by

raising the number to the power — возведением

числа

в степень

## Exercises

## Read the following words paying attention to the pronunciation:

inverse,

learn, perfect, order, for, opposite, not, must, number, thus.

to

square, to use, to raise, to multiply, to find, to check, to give, to

begin, to obtain, to get, to take, to be.

*Make*

up sentences of your own using the words and expressions given

below:

to

raise to power, to obtain the quantity, to square the number, to take

an arithmetic square root, to use a table may be checked, conclude.

1.

What operation should be performed to square a number? 2. What is a

perfect square? 3. What do we do to get the square root of a number?

4. What is the process of finding a root called? 5. How do we check

the accuracy of a root?

**V.**

Translate into Russian:

Tables

of squares are used by architects and engineers in working with

squares of number. If you have a table of squares, you can find the

approximate square root of any number. Sometimes it is not easy to

find a square root by

inspection.

If a table of squares is not at hand another method may be used.

Чтобы

возвести в квадрат число, надо умножить

это число на самое себя. Извлечение

квадратного корня — это действие

обратное возведению в квадрат. Чтобы

получить квадратный корень числа, мы

можем пользоваться специальной таблицей.

Правильность извлечения квадратного

корня можно проверить, возведя в квадрат

подкоренное

выражение;

если получится данное число, то корень

найден правильно.

**Asked by: Destiny Dicki**

A square root of a number is a value that, **when multiplied by itself, gives the number**. Example: 4 × 4 = 16, so a square root of 16 is 4. Note that (−4) × (−4) = 16 too, so −4 is also a square root of 16. The symbol is √ which always means the positive square root.

## What is the meaning of √ in maths?

**Square root**, in mathematics, a factor of a number that, when multiplied by itself, gives the original number.

## Where does square root come from?

**Chinese mathematical** writings from around 200BC show that square roots were being approximated using an excess and deficiency method. In 1450AD Regiomontanus invented a symbol for a square root, written as an elaborate R. The square root symbol √ was first used in print in 1525.

## How do you calculate square root?

The square root formula is used to find the square root of a number. We know the exponent formula: **n√x x n = x ^{1}^{/}^{n}**. When n= 2, we call it square root. We can use any of the above methods for finding the square root, such as prime factorization, long division, and so on.

## What is a square root of 144?

Thus the square root of 144 is **12**.

** 38 related questions found**

### What is the square root in simple terms?

A square root of a number is **a value that, when multiplied by itself, gives the number**. Example: 4 × 4 = 16, so a square root of 16 is 4. Note that (−4) × (−4) = 16 too, so −4 is also a square root of 16. The symbol is √ which always means the positive square root.

### Who invented math?

**Archimedes** is known as the Father of Mathematics. Mathematics is one of the ancient sciences developed in time immemorial.

### What you mean by root?

(Entry 1 of 5) 1a : **the usually underground part of a seed plant body that originates usually from the hypocotyl**, functions as an organ of absorption, aeration, and food storage or as a means of anchorage and support, and differs from a stem especially in lacking nodes, buds, and leaves.

### What is the answer for a square root called?

The term (or number) whose square root is being considered is known as **the radicand**. The radicand is the number or expression underneath the radical sign, in this case 9.

### Does a square root have two answers?

**It has multiple answers** so why do we pick the positive one? if x2=16⟹x=√16 or x=−√16 for respectively the positive and negative solution. This implies that the square root function has a single answer and we must negate its answer to obtain the second solution.

### IS 400 a perfect square?

The square root of a number is the number that when multiplied to itself gives the original number as the product. This shows that **400 is a perfect square**.

### Is 27 a cube number?

### Does 0 have a square root?

**Zero has one square root which is 0**. Negative numbers don’t have real square roots since a square is either positive or 0. The square roots of numbers that are not a perfect square are members of the irrational numbers. This means that they can’t be written as the quotient of two integers.

### Is 1 is a perfect square?

Informally: When you multiply an integer (a “whole” number, positive, negative or zero) times itself, the resulting product is called a square number, or a perfect square or simply “a square.” So, 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and so on, are all square numbers.

### What is the square root of 64?

The square root of 64 is **8**.

### What will be the unit digit of the square of 144?

144, 484 all are Perfect Square having unit **digit 4** and tens digit is even.

### Is 144 a cube number?

Is 144 a Perfect Cube? The number 144 on prime factorization gives 2 × 2 × 2 × 2 × 3 × 3. Here, the prime factor 2 is not in the power of 3. Therefore the cube root of 144 is irrational, hence **144 is not a perfect cube**.

Why is i the square root of negative one?, Here, the term “imaginary” is used because there is no real number having a **negative square**. There are two complex **square roots** of −**1**, namely i and −i, just as there are two complex **square roots** of every real number other than zero (which has **one** double **square root**).

Furthermore, What is the square root of a negative number?, Zero has one **square root** which is 0. **Negative numbers** don’t have real **square roots** since a **square** is either positive or 0. The **square roots** of **numbers** that are not a perfect **square** are members of the irrational **numbers**. This means that they can’t be written as the quotient of two integers.

Finally, What is the square of negative 3?, If you want to say “the **square of negative three**” you write (−**3**)2.

## Frequently Asked Question:

### Why is the square root of not a negative number?

Any **number** times itself is a positive **number** (or zero), so you can’t ever get to a **negative number** by squaring. Since **square roots** undo squaring, **negative numbers** can’t have **square roots**.

### Why can’t the square root of a number be negative?

### Why is square root only positive?

The reason for this distinction is that in a mathematical function f(x, y) for every value of x, there has to be a unique value of y. Thus, the **square root** of 4 cannot be +2, -2, by definition! Thus, as a norm, we **only** take the **square root** function to be **positive**.

### Can you have a negative number in a square root?

### Is a square root positive or negative?

Yes, it will always be the **positive** number. The point is that, by convention, **square root** refers to the principal **square root**. You say that **square root** only refers to the **positive**, yet for the first one you say it has 2 solutions.

### What is the square of negative 4?

Squaring **Negative Numbers**

For **example**, to square -4 enter it into the calculator as (-4) with parentheses. To take the negative of 4 squared enter it as -(4) or -4.

### What is a negative 3?

For example, **negative** three is the result of subtracting three from zero: 0 − **3** = −**3**. In general, the subtraction of a larger number from a smaller yields a **negative** result, with the magnitude of the result being the difference between the two numbers. For example, 5 − 8 = −**3**.

### Is the square root of negative 3 a real number?

### What is the square of 3?

**List of Perfect Squares**

### Can you find the square root of a negative number?

It is impossible to **find the square root** of **negative one**, or the **square root** of any **negative number**, because no **number** times itself **can** equal a **negative number**.

### What is the square of a negative number?

“This is because to square a number just means to multiply it by itself. For **example**, (−2) squared is (−2)(−2)=4. Note that this is positive because when you multiply two negative numbers you get a positive result.” – This, of course, is the exact opposite of what was asked, but it’s the given response.

### What is the square root of a negative 1?

The **square root** of **minus one** √(−**1**) is the “unit” Imaginary Number, the equivalent of **1** for Real Numbers. In mathematics the symbol for √(−**1**) is i for imaginary.

### What is a negative square root called?

Every positive real number has two **square roots**, one positive and one **negative**. For this reason, we use the radical sign √ to denote the principal (nonnegative) **square root**. and a **negative** sign in front of the radical −√ to denote the **negative square root**.

### How is the square root of negative 1?

### Can I square root a negative number?

It is not possible to **square** a **value** (multiply it times itself) and arrive at a **negative value**. So, what **do** we **do**? The **square root** of a **negative number** does not exist among the set of Real **Numbers**.

### What is the square root of I where I √ 1?

**What is the square root of i, where i=√**–**1** ? =**1√**2(**1**+i)or-**1√**2(**1**+i).

### Who invented the square root of negative one?

While **Girolamo Cardano** was working on solving cubic and quadratic equations in 1539, he encountered some formulas that involved square roots of negative numbers.

## Square Root

The square root of a number is the inverse operation of squaring a number. The square of a number is the value that is obtained when we multiply the number by itself, while the square root of a number is obtained by finding a number that when squared gives the original number.

If ‘a’ is the square root of ‘b’, it means that a × a = b. The square of any number is always a positive number, so every number has two square roots, one of a positive value, and one of a negative value. For example, both 2 and -2 are square roots of 4. However, in most places, only the positive value is written as the square root of a number.

## What is Square Root?

The **square root of a number** is that factor of a number which when multiplied by itself gives the original number. Squares and square roots are special exponents. Consider the number 9. When 3 is multiplied by itself, it gives 9 as the product. This can be written as 3 × 3 or 3^{2}. Here, the exponent is 2, and we call it a square. Now when the exponent is 1/2, it refers to the square root of the number. For example, √n = n^{1/2}, where n is a positive integer.

### Square Root Definition

The square root of a number is the value of power 1/2 of that number. In other words, it is the number whose product by itself gives the original number. It is represented using the symbol ‘√ ‘. The square root symbol is called a radical, whereas the number under the square root symbol is called the radicand.

## How to Find Square Root?

- Repeated Subtraction Method
- Prime Factorization Method
- Estimation Method
- Long Division Method

It should be noted that the first three methods can be conveniently used for perfect squares, while the fourth method, i.e., the long division method can be used for any number whether it is a perfect square or not.

### Repeated Subtraction Method of Square Root

This is a very simple method. We subtract the consecutive odd numbers from the number for which we are finding the square root, till we reach 0. The number of times we subtract is the square root of the given number. This method works only for perfect square numbers. Let us find the square root of 16 using this method.

- 16 — 1 = 15
- 15 — 3 =12
- 12 — 5 = 7
- 7- 7 = 0

You can observe that we have subtracted 4 times. Thus,√16 = 4

### Square Root by Prime Factorization Method

**Step 1:**Divide the given number into its prime factors.**Step 2:**Form pairs of factors such that both factors in each pair are equal.**Step 3:**Take one factor from the pair.**Step 4:**Find the product of the factors obtained by taking one factor from each pair.**Step 5:**That product is the square root of the given number.

Let us find the square root of 144 by this method.

This method works when the given number is a perfect square number.

### Finding Square Root by Estimation Method

Estimation and approximation refer to a reasonable guess of the actual value to make calculations easier and more realistic. This method helps in estimating and approximating the square root of a given number. Let us use this method to find √15. Find the nearest perfect square number to 15. 9 and 16 are the perfect square numbers nearest to 15. We know that √16 = 4 and √9 = 3. This implies that √15 lies between 3 and 4. Now, we need to see if √15 is closer to 3 or 4. Let us consider 3.5 and 4. Since 3.5^{2} = 12.25 and 4^{2}= 16. Thus, √15 lies between 3.5 and 4 and is closer to 4.

Let us find the squares of 3.8 and 3.9. Since 3.8^{2} = 14.44 and 3.9^{2} = 15.21. This implies that √15 lies between 3.8 and 3.9. We can repeat the process and check between 3.85 and 3.9. We can observe that √15 = 3.872.

This is a very long process and time-consuming.

### Calculating Square Root by Long Division Method

Long division is a method for dividing large numbers into steps or parts, breaking the division problem into a sequence of easier steps. We can find the exact square root of any given number using this method. Let us understand the process of finding square root by the long division method with an example. Let us find the square root of 180.

**Step 1:**Place a bar over every pair of digits of the number starting from the units’ place (right-most side). We will have two pairs, i.e., 1 and 80**Step 2:**We divide the left-most number by the largest number whose square is less than or equal to the number in the left-most pair.

**Step 3:** Bring down the number under the next bar to the right of the remainder. Add the last digit of the quotient to the divisor. To the right of the obtained sum, find a suitable number which, together with the result of the sum, forms a new divisor for the new dividend that is carried down.

**Step 4:** The new number in the quotient will have the same number as selected in the divisor. The condition is the same — as being either less than or equal to the dividend.

**Step 5:** Now, we will continue this process further using a decimal point and adding zeros in pairs to the remainder.

## Square Root Table

The square root table consists of numbers and their square roots. It is useful to find the squares of numbers as well. Here is the list of square roots of perfect square numbers and some non-perfect square numbers from 1 to 10.

The numbers that are not perfect squares are irrational numbers.

## Square Root Formula

## Simplifying Square Root

For fractions, there is also a similar rule: √x/√y = √(x/y). For example: √50/√10 = √(50/10)= √5

## Square Root of a Negative Number

The square root of a negative number cannot be a real number, since a square is either a positive number or zero. But complex numbers have the solutions to the square root of a negative number. The principal square root of -x is: √(-x)= i√x. Here, i is the square root of -1.

For example: Take a perfect square number like 16. Now, let’s see the square root of -16. There is no real square root of -16. √(-16)= √16 × √(-1) = 4i (as, √(-1)= i), where ‘i’ is represented as the square root of -1. So, 4i is a square root of -16.

## Square of a Number

Any number raised to exponent two (y^{2}) is called the square of the base. So, 5^{2} or 25 is referred to as the square of 5, while 8^{2} or 64 is referred to as the square of 8. We can easily find the square of a number by multiplying the number two times. For example, 5^{2} = 5 × 5 = 25, and 8^{2} = 8 × 8 = 64. When we find the square of a whole number, the resultant number is a perfect square. Some of the perfect squares we have are 4, 9, 16, 25, 36, 49, 64, and so on. The square of a number is always a positive number.

## How to Find the Square of a Number?

The square of a number can be found by multiplying a number by itself. For single-digit numbers, we can use multiplication tables to find the square, while in the case of two or more than two-digit numbers, we perform multiplication of the number by itself to get the answer. For example, 9 × 9 = 81, where 81 is the square of 9. Similarly, 3 × 3 = 9, where 9 is the square of 3.

The square of a number is written by raising the exponent to 2. For example, the square of 3 is written as 3^{2} and is read as «3 squared». Here are some examples:

- 4
^{2}= 4 × 4 = 16 - (-6)
^{2}= -6 × -6 = 36 - (5/3)
^{2}= 5/3 × 5/3 = 25/9

## Squares and Square Roots

There is very strong relation between squares and square roots as each one of them is the inverse relation of the other. i.e., if x^{2} = y then x = √y. It can be simply remembered like this:

- When «square» is removed from one side of the equation, we get the square root on the other side. For example, 4
^{2}= 16 means, 4 = √16. This is also known as «taking square root on both sides». - When «square root» is removed from one side of the equation, we get square on the other side. For example, √25 = 5 means, 25 = 5
^{2}. This is also known as «squaring on both sides»

**Example:** Solve the equation √(2x + 3) = 10.

Squaring on both sides of the equation would result in the cancellation of the square root on the left side.

2x + 3 = 10^{2}

2x + 3 = 100

2x = 97

x = 97/2 = 48.5

Here are more differences between squares and square roots.

☛ **Related Articles**

## Square Root of Numbers

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## FAQs on Square Root

### What is Square Root in Math?

The **square root** of a number is a number that when multiplied by itself gives the actual number. For example, 2 is the square root of 4, and this is expressed as √4 = 2. This means when 2 is multiplied by 2 it results in 4 and this can be verified as 2 × 2 = 4.

### How to Calculate the Square Root of a Number?

It is very easy to find the square root of a number that is a perfect square. For example, 9 is a perfect square, 9 = 3 × 3. So, 3 is the square root of 9 and this can be expressed as √9 = 3. The square root of any number, in general, can be found by using any of the four methods given below:

- Repeated Subtraction Method
- Prime Factorization Method
- Estimation and Approximation Method
- Long Division Method

### Can Square Root be Negative?

Yes, the square root of a number can be negative. In fact, all the perfect squares like 4, 9, 25, 36, etc. have two square roots, one is a positive value and one is a negative value. For example, the square roots of 4 are -2 and 2. To verify this, we can see that (-2) × (-2) = 4. Similarly, the square roots of 9 are 3 and -3.

### How to Find the Square Root of a Decimal Number?

The square root of a decimal number can be found by using the estimation method or the long division method. In the case of decimal numbers, we make pairs of whole number parts and fractional parts separately. And then, we carry out the process of long division in the same way as any other whole number.

### What is the Square Root Symbol?

The symbol that is used to denote square root is called the radical sign ‘√ ‘. The term written inside the radical sign is called the radicand.

### How to Multiply Two Square Root Values Together?

Let us say we have two numbers a and b. First, we will find the square root of the numbers a and b. Then, after finding the square root we will multiply the square roots value together. Let us understand this with a practical illustration. For example, multiply √4 × √16. The square root of 4 is 2 (√4 = 2) and the square root of 16 is 4 (√16 = 4). Now, we will multiply the value of the square root of 4 and 16, i.e., 2 × 4 = 8. Instead, we can apply the property of square roots, √a × √b = √ab.

### What is the Formula for Calculating the Square Root of a Number?

The square root of any number can be expressed using the formula: √y = y^{½}. In other words, if a number has 1/2 as its exponent, it means we need to find the square root of the number.

### What is the Square and Square Root of a Number?

The square of a number is the product that we get on multiplying a number by itself. For example, 6 × 6 = 36. Here, 36 is the square of 6. The square root of a number is that factor of the number and when it is multiplied by itself the result is the original number. Now, if we want to find the square root of 36, that is, √36, we get the answer as, √36 = 6. Hence, we can see that the square and the square root of a number are inverse operations of each other.

### Which Method is Used to Find the Square Root of Non-Perfect Square Numbers?

In Math, a non-perfect or an imperfect square number is considered as a number whose square root cannot be found as an integer or as a fraction of integers. The square root of a non-perfect square number can be calculated by using the long division method.

### How to Find a Square Root on a Calculator?

To find the square root value of any number on a calculator, we simply need to type the number for which we want the square root and then insert the square root symbol √ in the calculator. For example, if we need to find the square root of 81, we should type 81 in the calculator and then press the symbol √ to get its square root. We will get √81 = 9.

### What are the Applications of the Square Root Formula?

There are various applications of the square root formula:

- The square root formula is mainly used in algebra and geometry.
- It helps in finding the roots of a quadratic equation.
- It is widely used by engineers.

### What does the Square of a Number mean?

The product that we get on multiplying a number by itself is the square of the number. For example, 5 × 5 = 25. Here, 25 is the square of 5 and this can also be written as 5^{2} = 25.

### How to Calculate the Square Root of a Negative Number?

Note that the square root of a negative number is not a real number. It is an imaginary number. For example, √(-4) = √(-1) × √4 = i (2) = 2i, where ‘i’ is known as «iota» and i^{2} = -1 (or) i = √(-1).

### Why is the Square of a Negative Number Positive?

The square of a negative number is positive because when two negative numbers are multiplied it always results in a positive number. For example, (-4) × (-4) = 16.

## Square Root of 1

The square root of 1 is expressed as √1 in the radical form and as (1)^{½} or (1)^{0.5} in the exponent form. It is the positive solution of the equation x^{2} = 1.

**Square Root of 1:**1**Square Root of 1 in exponential form:**(1)^{½}or (1)^{0.5}**Square Root of 1 in radical form:**√1

## What Is the Square Root of 1?

The square root is an inverse mathematical operation of a square. As you are aware, exceptional cases are always there. In mathematics, there are too many exceptions we deal with. Look at the multiplication shown below when 1 is multiplied by 1 we get a new number as 1, 1 is a square root of 1, or square of 1 is also 1. This is an exceptional case of a square root of 1.

## Is Square Root of 1 Rational or Irrational?

Since √1** = **1 which is rational numbers. Hence, the square root of 1 is rational.

## How to Find the Square Root of 1?

Explore some square roots of other numbers using illustrations and interactive examples.

- Square root 5
- Square root 2
- Square root 4
- Square root 11
- Square root 9

- Can you think what will be the square root of -11 and -111?

- 1 is the only number whose square and the square root are equal.
- 1 is neither prime nor composite.

## Square Root of 1 Solved Examples

Find out the perfect square numbers between 1 to 5. Are the numbers rational or irrational?

The square root of 1 = 1

The square root of 2 = 1.414

The square root of 3 = 1.732

The square root of 4 = 2

The square root of 5 = 2.236

The perfect square numbers between 1 to 5 are 1 and 4.

The square root of 1 and 4 is a rational number.**Example 2**Jacob is trying different methods to find the square root of a number. Help him find √1 by prime factorization method.Prime factorization of 1 = 1 × 1

Now, the square root of 1 would be the product of 1 digit from each pair.

Therefore, √1 = 1**Example**If the surface area of a sphere is 4π in^{2}. Find the radius of the sphere.Let ‘r’ be the radius of the sphere.

⇒ Area of the sphere = 4πr^{2}= 4π in^{2}

⇒ r = ±√1 in

Since radius can’t be negative,

⇒ r = √1

The square root of 1 is 1.

⇒ r = 1 in

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## FAQs on the Square Root of 1

### What is the Value of the Square Root of 1?

The square root of 1 is 1.

### If the Square Root of 1 is 1. Find the Value of the Square Root of 0.01.

Let us represent √0.01 in p/q form i.e. √(1/100) = 0.01/10 = 0.1. Hence, the value of √0.01 = 0.1

### What is the Square Root of -1?

The square root of -1 is an imaginary number. It is also written as √-1 = i

where i = √-1 and it is called the imaginary unit.

Algebra is an important topic of mathematics. A square root is an operation that is used in many formulas and different fields of mathematics. This article is about square root and square root formula. The square root of a number is a number squaring which gives the original number. There are multiple square root formulas that are discussed in this article with their problems.

### Square Root

The square root of a number is a number squaring which gives the original number. It is that factor of the number that when squared gives the original number. It is the value of power 1/2 of that number. The square root of a number is represented as** √**.

**Example:** Square root of 9 ⇒ √9 = ± 3

Here, 3^{2 }= 9

(-3)^{2 }= 9

**The number inside the square root is called radicand and the square root symbol is called radical.**

**Methods for finding the square root of a number**

There are two methods for finding the square roots of a number that are highly used in mathematics. These are discussed below.

- Prime Factorization Method
- Long Division Method

### Prime Factorization Method

Prime factorization is a method in which a number is represented in the form of the product of prime numbers. Then the square root of the number is fined according to the given concept.

**To find the square root using the prime factorization method:**

Step 1:Represent the number in its prime factors using prime factorization method.

Step 2:Form the pair of the same factors.

Step 3:Take one factor from each pair and then, find the products of all the factors obtained by taking one factor from each pair.

Step 4:The resultant product is the square root of the number.

**Example for Prime Factorization Method:**

### Long Division Method

Steps to find the square root of a number using the long division method:

**Example: Find the square root of 256 using the long division method.**

**Step 1.** Divide the number into pairs starting from one place. **For example, pairs starting from one place:= 2, 56**

**Step 2.** After dividing the digits into pairs, start from the leftmost pair or digit. The largest number whose square is just less than or equal to the first pair or digit is taken as the divisor and also the quotient. **In the above example, the largest number whose square is just less than 2 is 1. So, the divisor is 1 and the quotient is also 1.**

**Step 3. **Subtract the square of the divisor from the first pair or digit and bring the next pair down to the right of the reminder to get the new dividend.** In the above example 2 – 1 = 1 then, we bring the next pair i.e. 56 down and the new dividend becomes 156.**

**Step 4.** Now, the new divisor is obtained by adding the previous divisor and the previous quotient digit and concatenating it with a suitable digit which is also taken as the next digit of the quotient, chosen in a way that the product of the new divisor and this new digit in quotient is equal to or just less than the new dividend. **In the above example, the previous divisor is 1 and the previous quotient digit is 1 and their addition gives 2 which is the new divisor. Now, we have to choose a digit so that the product of the new divisor and the new digit in the quotient is equal to or less than the new dividend i.e. 26 is the new divisor and 6 in the new digit which is concatenated with the previous quotient. Now, the current quotient is 16.**

**Step 5.** Repeat steps 2, 3, and 4 till all the pairs have been taken. Now, the resultant quotient is the square root of the given number.** In the above example, all pairs have been taken, and hence, the square root of the number 256 is 16. **

### Table for Square and Square root of first ten natural numbers

### Square Root Formula

The square root of a number has the exponent 1/2. The square root formula is used to find the square root of a number.

**Square Root Formula : √x = x ^{1/2}**

### Sample Problems

**Question 1: Find the sum: 5√3 + 6√12**

5√3 + 6√12 = 5√3 + 6(√(4 × 3)= 5√3 + 6 × √4 × √3

= 5√3 + 6 ×2√3

= 5√3 + 12√3

**Question 2: Evaluate: √64 – √25**

√64 – √25 = √(8 × 8) – √(5 × 5 )

= 8 – 5 = 3

**Question 3: Evaluate: √63 / √28**

√63 / √28 = √(7 × 9 )/ √(7 × 4 )

= √(9 /4)

= √9 / √4

**Question 4: Evaluate: 5 /√15 **

5 /√15 = (5 /√15)×(√15 /√15 )

**Question 5: Evaluate: 4 / (5 + √6)**

= (20 – 4√6) / (25 – 6)

= (20 – 4√6) / 19

**Question 6: Evaluate : 7 / (8 – √10)**

= (56 + 7√10) / (64 – 10)

= (56 + 7√10) / 54

**Question 7: Evaluate : (2 +√5) / (4 – √2)**

The arithmetic value which is used for representing the quantity and used in making calculations are defined as Numbers. A symbol like “4,5,6” which represents a number is known as a numeral. Without numbers, we can’t do counting of things, date, time, money, etc., these numbers are also used for measurement and used for labeling.

The properties of numbers make them helpful in performing arithmetic operations on them. These numbers can be written in numeric forms and also in words.

There are different types of numbers, which we can learn. They are whole and natural numbers, odd and even numbers, rational and irrational numbers, etc.

**What is a Number System?**

A Number System is a method of showing numbers by writing, which is a mathematical way of representing the numbers of a given set, by using the numbers or symbols in a mathematical manner. The writing system for denoting numbers using digits or symbols in a logical manner is defined as Number System.

We can use the digits from 0 to 9 to form all the numbers. With these digits, anyone can create infinite numbers.

For example, 156, 3907, 3456, 1298, 784859, etc.

**What is a Square Root?**

The value of a number of square roots, which on multiplication by itself gives the original number. Suppose, a is the square root of b, then it is represented as a = √b or we can express the same equation as **a ^{2} = b.** Here, ’√’ this symbol we used to represent the root of numbers is termed as radical. The positive number when it is to be multiplied by itself represents the square of the number. The square root of the square of any positive number gives the original number.

**For example,** the square of 4 is 16, 4^{2} = 16, and the square root of 16, **√16 = ±4. **Since 4 is a perfect square, hence it is easy to find the square root of such numbers, but for an imperfect square, it’s really tricky.

Square Root is represented as** ‘√’.** It is called a radical symbol. To represent a number ‘a’ as a square root using this symbol can be written as: ‘√a‘, where a is the number.

The number here under the radical symbol is called the radicand. For example, the square root of 4 is also represented as a radical of 4. Both represent the same value.

The formula to find the square root is: **a = √b**

**Properties of Square Roots**

It is defined as a one-to-one function that takes a positive number as an input and returns the square root of the given input number.

**f(x) = √x**

For example, here if x = 9, then the function returns the output value as 3.

- If a number is a perfect square number, then there definitely exists a perfect square root.
- If a number ends with an even number of zeros (0’s), then we can have a square root.
- The two square root values can be multiplied. For example, √3 can be multiplied by √2, then the result will be √6.
- When two same square roots are multiplied, then the result must be a radical number. It shows that the result is a non-square root number. For example, when √7 is multiplied by √7, the result obtained is 7.
- The square root of negative numbers is undefined. Hence the perfect square cannot be negative.
- Some of the numbers end with 2, 3, 7, or 8 (in the unit digit), then the perfect square root does not exist.
- Some of the numbers end with 1, 4, 5, 6, or 9 in the unit digit, then the number will have a square root.

It is easy to find the square root of a number that is a perfect square.

**Perfect squares** are those positive numbers that can be written as the multiplication of a number by itself, or you can say that a perfect square is a number which is the value of power 2 of any integer.

The number that can be expressed as the product of two equal integers. For example, 16 is a perfect square because it is the product of two equal integers, 4 × 4 = 16. However, 24 is not a perfect square because it cannot be expressed as the product of two equal integers. (8 × 3 = 24).

The number which is obtained by squaring a whole number is termed as a perfect square. If we assume N is a perfect square of a whole number y, this can be written as N = the product of y and y = y

^{2}.

So, the perfect square formula can be expressed as:

N = Y^{2}Let’s Use the formula with values.

If y = 9, and N = y

^{2}.This means, N = 9

^{2}= 81.Here, 81 is a perfect square of 9 because it is the square of a whole number.

So real square roots of 81 is +9, -9

With the help of square roots, we can identify whether a number is a perfect square or not, if we calculate the square root of the given number.

If the square root is a whole number then the given number will be a perfect square, and if the square root value is not a whole number, then the given number is not a perfect square.

For instance,to check whether 24 is a perfect square or not, we will calculate its square root. √24 = 4.898979. As we can see, 4.898979 is not a whole number, so, 24 is not a perfect square.

Let’s take another example of

The number 49.

√49 = ±7.We can see that 7 is a whole number, therefore, 49 is a perfect square.

### Why do we use plus or minus in square root?

If we want both the positive and the negative square root of a radicand then we put the symbol ± (read as plus minus) in front of the root.

The numbers that are not a perfect square are members of the irrational numbers. This means that numbers or square root can’t be written as the quotient of two integers.

### Sample Problems

**Question 1: What are the two square roots of 100?**

Here 100 is the perfect square of 10, so this can have two roots one negative and one positive

or we can say real square root of 100 is ±10

or 10

^{2}= 10 × 10 = 100(-10)

^{2}= – 10 × – 10 = 100

Hence, the two square roots of 100 are +10 and -10.

**Question 2: What are the square roots of 12?**

Square root of 12

Here 12 is not a perfect square so this number doesn’t have two square roots we can’t write it as √12 = ±3.464

Therefore √12 = 3.464 is an irrational number, the numbers that are not a perfect square are members of the irrational numbers. This means that numbers or square roots can’t be written as the quotient of two integers.

**Question 3: What are the two square roots of 144? **

square root of 144

Here square root of 144 is perfect square of 12, i.e a whole number this has two square roots +12, -12

Therefore

√144 = ± 12