Practice some problems before going into the exercise.
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 Calculator, Practice Problems, and Answers
 Things You’ll Need
 About This Article
 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
 Solving Quadratic Equations: Video Lesson
 Quadratic Equation Solver
 How to Solve Quadratic Equations using Square Roots
 How to Solve Quadratic Equations by Factoring
 Solving Quadratic Equations by Factoring: Example 1
 Solving Quadratic Equations by Factoring: Examples with Negatives
 How to Solve Quadratic Equations with Difference of Two Squares
 How to Solve Quadratic Equations with a Coefficient of 𝑥2 Greater than 1
 Examples of Solving Quadratic Equations with a>1 by Factoring
 How to Solve Quadratic Equations by Completing the Square
 Example of Solving a Quadratic Equation with a Leading Coefficient Greater than 1 using Completing the Square
 Examples of Solving Quadratic Equations by Completing the Square
 How to Solve Quadratic Equations with the Quadratic Formula
 Solving Quadratic Equations with the Quadratic Formula Examples
 Solving Quadratic Equations Not Written in Standard Form
 Solving a Quadratic Equation Not Written in Standard Form: Example 1
 Solving a Quadratic Equation Not Written in Standard Form: Example 2
 Solving a Quadratic Equation when given ‘y’
 How to Solve Quadratic Equations by Graphing
 Solving Quadratic Equations with Complex Solutions
 Преобразование выражений, содержащих квадратный корень
 Notes:
 Square Root:
 Square Root Symbol / Square Root Sign:
 How are square roots used:
 How to Calculate the Square Root of:
 Need to calculate the square root of a number with a handheld calculator or looking to buy a new calculator?
 Graphing Calculator:
 Texas Instruments TI84 Plus Graphing Calculator
 Casio Graphing Calculator (fx9750GII)
 Scientific Calculator:
 Texas Instruments TI36X Pro Scientific Calculator
 Math Help / Math Tutor
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You can multiply square roots, a type of radical expression, just as you might multiply whole numbers. Sometimes square roots have coefficients (an integer in front of the radical sign), but this only adds a step to the multiplication and does not change the process. The trickiest part of multiplying square roots is simplifying the expression to reach your final answer, but even this step is easy if you know your perfect squares.

Multiply the coefficients. A coefficient is a number in front of the radical sign. To do this, just ignore the radical sign and radicand, and multiply the two whole numbers. Place their product in front of the first radical sign.



Calculator, Practice Problems, and Answers
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We are not allowed to use a calculator, so how do I multiply a whole number by a square root?
When you multiply a whole number by a square root, you just put the two together, with the whole number in front of the square root. For example, 2 * (square root of 3) = 2(square root of 3). If the square root has a whole number in front of it, multiply the whole numbers together. So 2 * 4(square root of 3) = 8(square root of 3).
What is 2 root 3 times root 3?
√3 times √3 equals 3. Two times that is 6.
What is 4 divided by square root of 5?
(4√5)/5. Since radicals are not supposed to be in the denominator, you multiply by √5/√5 to get (4√5)/5.
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Always remember your perfect squares because it will make the process much easier!
All terms under the radicand are always positive, so you will not have to worry about sign rules when multiplying radicands.
Things You’ll Need
 Pencil
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 Calculator
About This Article
To multiply square roots, first multiply the radicands, or the numbers underneath the radical sign. If there are any coefficients in front of the radical sign, multiply them together as well. Finally, if the new radicand can be divided out by a perfect square, factor out this perfect square and simplify it. If you want to learn how to check your answers when you’re finished solving, keep reading the article!
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Do you want to learn how to solve some square root math problems? Here are some examples of square root problems that we have explained and solved:
Unit digit of the square of 272
Square Root of Pi
10000 square feet to acres
What is the square root of 3?
What is the square root of 5 squared?
What is the square root of 1/4?
What is sum of the first 10 square numbers?
Is the square root of 40 rational or irrational?
Simplify Square Root of 72
How to find the square root of 80 by hand?
What is the square root of 50 as a fraction?
Where can I get a list of perfect squares?
What is the square root of negative 4?
How to convert 200 square meters to square feet
Convert 2 acres to square feet
How many square feet is a 12×12 room?
What is Square Root of 0.25?
1 square foot to square meters
Square root of 2 definition
The square root of 2 in mathematical form is written with the radical sign like this √2. We call this the square root of 2 in radical form.
The square root of 2 is a quantity (q) that when multiplied by itself will equal 2.
√ = q × q = q^{2}
Is 2 a perfect square?
2 is a perfect square if the square root of 2 equals a whole number. As we have calculated further
down on this page, the square root of 2 is not a whole number.
2 is not a perfect square.
Is the square root of 2 rational or irrational?
The square root of 2 is a rational number if 2 is a perfect square. It is an irrational number if it is not a perfect square.
Since 2 is not a perfect square, it is an irrational number. This means that the answer to «the square root of 2?» will have an infinite number
of decimals. The decimals will not terminate and you cannot make it into an exact fraction.
√ is an irrational number
Can the square root of 2 be simplified?
You can simplify 2 if you can make 2 inside the radical smaller. We call this process «to simplify a surd».
The square root of 2 cannot be simplified.
√ is already in its simplest radical form.
√ ≈ 1.414213562
How to calculate the square root of 2 with a computer
If you are using a computer that has Excel or Numbers, then you can enter SQRT(2) in a cell to get the square root of 2.
Below is the result we got with 13 decimals. We call this the square root of 2 in decimal form.
SQRT(2) ≈ 1.4142135623731
What is the square root of 2 rounded?
The square root of 2 rounded to the nearest tenth, means that you want one digit after the decimal point. The square root of 2 rounded to the nearest hundredth, means that you
want two digits after the decimal point. The square root of 2 rounded to the nearest thousandth, means that you want three digits after the decimal point.
10th: √ ≈ 1.4
100th: √ ≈ 1.41
1000th: √ ≈ 1.414
What is the square root of 2 as a fraction?
Like we said above, since the square root of 2 is an irrational number, we cannot make it into an exact fraction.
However, we can make it into an approximate fraction using the square root of 2 rounded to the nearest hundredth.
√
≈ 1.41/1
≈ 141/100
≈ 1 41/100
What is the square root of 2 written with an exponent?
All square roots can be converted to a number (base) with a fractional exponent. The square root of 2 is no exception. Here is the rule and the answer
to «the square root of 2 converted to a base with an exponent?»:
√ = b^{½}
√ = 2^{½}
How to find the square root of 2 by long division method
Here we will show you how to calculate the square root of 2 using the long division method with one decimal place accuracy. This is the lost
art of how they calculated the square root of 2 by hand before modern technology was invented.
Set up 2 in pairs of two digits from right to left and attach one set of 00 because we want one decimal:
Starting with the first set: the largest perfect square less than or equal to 2 is 1, and the square root of 1 is 1. Therefore, put 1 on top and 1 at the bottom like this:
Calculate 2 minus 1 and put the difference below. Then move down the next set of numbers.
Double the number in green on top: 1 × 2 = 2. Then, use 2 and the bottom number to make this problem:
2 × ≤ 100
The question marks are «blank» and the same «blank». With trial and error, we found the largest number «blank» can be is 4.
Now, enter 4 on top:
That’s it! The answer is on top. The square root of 2 with one digit decimal accuracy is 1.4.
Square Root of a Number
Please enter another number in the box below to get the square root of the number and other detailed information like you got for 2 on this page.
Remember that negative times negative equals positive. Thus, the square root of 2 does not only have the positive answer
that we have explained above, but also the negative counterpart.
We often refer to perfect square roots on this page. You may want to use the list of perfect squares
for reference.
Square Root of 3
Here is the next number on our list that we have equally detailed square root information about.
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 onehalf:
but only for x greater than or equal to 0
Solving Quadratic Equations: Video Lesson
Quadratic Equation Solver
A quadratic equation of the form can be solved using the quadratic formula where:
 ‘a’ is the coefficient of of 𝑥^{2}.
 ‘b’ is the coefficient of 𝑥.
 ‘c’ is the constant term
Enter the values of ‘a’, ‘b’ and ‘c’ into the quadratic equation solver below to calculate the values of 𝑥 that satisfy the equation.
For example, the quadratic equation has ‘a’=2, ‘b’=1 and ‘c’=3.
To solve a quadratic equation, the first step is to set one side of the equation equal to zero. The 3 ways to solve quadratic equations are factoring, completing the square or using the quadratic formula. If a quadratic equation cannot be factorised then completing the square or quadratic formula must be used.
If the quadratic has an 𝑥^{2} coefficient greater than 1 or cannot be factorised, the quadratic formula can be used:
Here is a list of the methods that can be used to solve quadratic equations:
 If 𝑥^{2} equals a number, square root both sides of the equation to solve it.
 If the quadratic only contains 𝑥^{2} and 𝑥 terms, factorise the 𝑥 out and solve.
 Try to factorise by finding two numbers that add to make the coefficient of 𝑥 and multiply to make the constant term. Solve by setting each factor to equal zero.
 If the quadratic contains an 𝑥^{2} coefficient greater than 1, try to split the 𝑥 term and factorise by grouping.
 If the quadratic cannot be factorised, complete the square and solve.
 If the quadratic cannot be factorised, use the quadratic formula .
All quadratic equations can be solved using the quadratic formula so this method will always work for solving quadratic equations.
How to Solve Quadratic Equations using Square Roots
If a quadratic equation is of the form 𝑥^{2}=k, square root both sides. The solutions are 𝑥=±√k. For example, if 𝑥^{2}=4 then 𝑥=±2.
This simple type of quadratic equation can be identified as there is only an 𝑥^{2} and constant term in the equation.
In this next example, 𝑥^{2} is equal to a non square number. We still square root both sides of the equation to obtain the solution.
For example, solve 𝑥^{2}5=0.
The first step is to add 5 to both sides of the equation so that 𝑥^{2}=5.
The next step is to square root both sides of the equation so that 𝑥=±√5.
Evaluating ±√5 on a calculator, 𝑥≈2.24 or 𝑥≈2.24.
How to Solve Quadratic Equations by Factoring
To solve a quadratic equation ‘𝑥^{2}+b𝑥+c=0′ by factoring:
 Think of two numbers that add to make b and multiply to make c.
 Factor the quadratic as (𝑥+m)(𝑥+n)=0, where m and n are the two numbers from step 1.
 Solve the quadratic by setting each bracket equal to zero.
 The solutions are 𝑥=m and 𝑥=n.
Solving Quadratic Equations by Factoring: Example 1
For example, solve the quadratic equation by factoring.
Here, and .
Step 1. Think of two numbers that add to make b and multiply to make c
The numbers 1 and 3 add to make 4 and multiply to make 3.
Step 2. Factor the quadratic as (𝑥+m)(𝑥+n)=0, where m and n are the two numbers from step 1
In step 1, the two numbers that add to make 4 and multiply to make 3 were 1 and 3.
Therefore the quadratic equation can be factorised to .
Step 3. Solve the quadratic by setting each bracket equal to zero
Either or .
We find the values of 𝑥 that make each bracket equal zero.
For , the solution is .
For the solution is .
Therefore the solutions to this quadratic equation are 𝑥=3 and 𝑥=1.
Solving Quadratic Equations by Factoring: Examples with Negatives
To solve a quadratic by factoring, think of two numbers that add to make the coefficient of 𝑥 and multiply to make the constant term.
 If b and c are both positive, the two numbers will both be positive
 If c is positive but b is negative, the two numbers will both be negative
 If c is negative, one number will be positive and one number will be negative
It is easier to factorise quadratics if you know what signs to expect in the brackets.
Here are some examples of solving quadratic equations by factoring.
 In example 1, .
Since b and c are positive, the quadratic will factor as .
The two numbers that add to make 6 and multiply to make 8 are 4 and 2.
Therefore, the factored quadratic is and so, and .
 In example 2, .
Here, c is positive and b is negative so the quadratic will factor as .
The two numbers that add to make 5 and multiply to make 6 are 3 and 2.
Therefore the factored quadratic is and so, and .
 In example 3, .
Here c is negative so the quadratic will factor as .
The two numbers that add to make 2 and multiply to make 3 are 3 and 1.
Therefore the factored quadratic is and so, or .
 In example 4, .
Here c is negative so the quadratic will factor as .
The two numbers that add to make 3 and multiply to make 10 are 2 and 5.
Therefore the factored quadratic is and so, and .
How to Solve Quadratic Equations with Difference of Two Squares
Any quadratic equation of the form 𝑥^{2}a^{2}=0 can be factored as (𝑥+a)(𝑥a)=0 using the difference of two squares. Therefore the solutions are 𝑥=a and 𝑥=a. For example, 𝑥^{2}25=0 factors as (𝑥+5)(𝑥5)=0 and has solutions of 𝑥=5 and 𝑥=5.
Quadratic equations of the form can be solved using the difference of two squares method.
To identify the difference of two squares, the quadratic should be made up of two terms separated by a minus sign.
For the quadratic , this is the same as .
Therefore this can be factored as and setting each bracket equal to zero to solve, and .
The difference of two squares can also be used to solve quadratics of the form . These quadratics will factor in the form which can be solved by setting each bracket equal to zero.
For example, solve using the difference of two squares.
This can be factorised as .
The solutions are therefore and .
How to Solve Quadratic Equations with a Coefficient of 𝑥^{2} Greater than 1
To solve a quadratic equation with a leading coefficient greater than 1, use the quadratic formula or factorise and solve.
To factor a quadratic a𝑥^{2}+b𝑥+c when a is greater than 1:
 Split the middle term into two terms that multiply to make the value of a×c.
 Factor the first two terms and the last two terms separately.
 Factor the whole expression by grouping.
For example, factor and solve the quadratic .
In this quadratic, , and
Step 1. Split the middle term into two terms that multiply to make the value of a×c
The middle term of is . This will be split into two terms that when multiplied, equal the value of 2×6.
is split into because 4×3=12, which is the same as 2×6.
Step 2. Factor the first two terms and the last two terms separately
The quadratic is currently written as .
The first two terms can be factorised as and the last two terms can be factorised as .
Therefore can be written as .
Step 3. Factor the whole expression by grouping
In the equation , there is a factor in common which is .
can be grouped to get .
Step 4. Solve the quadratic by setting each bracket equal to zero
The quadratic is currently written as .
Therefore either or .
Solving these equations, or .
Examples of Solving Quadratic Equations with a>1 by Factoring
Here are some examples of solving quadratic equations with a leading coefficient greater than 1 using the factorising by grouping method.
How to Solve Quadratic Equations by Completing the Square
All quadratic equations can be solved by completing the square and then rearranging the equation for 𝑥. For example, the equation 𝑥^{2}+4𝑥+1=0 can be written as (𝑥+2)^{2}3=0. Adding 3 to both sides of the equation, (𝑥+2)^{2}=3. Taking the square root, 𝑥+2=±√3 and so, 𝑥=2±√3.
 Complete the square, writing 𝑥^{2}+b𝑥+c=0 in the form
 Move the constant term to the other side of the equals sign
 Square root both sides
 Solve for 𝑥
For example, complete the square and solve, .
Step 1. Complete the square
To complete the square, write in the form .
In , and .
Therefore completing the square using the above formula, becomes .
This simplifies to .
Completing the square has now been completed and now we solve the equation by rearranging for 𝑥.
Step 2. Move the constant term to the other side of the equals sign
We add 3 to both sides of the equation so that becomes
Step 3. Square root both sides
Taking the square root of both sides, becomes .
Remember, when taking the square root, there are two solutions. A positive and a negative solution.
Step 4. Solve for 𝑥
Simply subtract two from both sides so that becomes .
The two solutions are and .
Example of Solving a Quadratic Equation with a Leading Coefficient Greater than 1 using Completing the Square
If a quadratic equation has a leading coefficient greater than 1, first factor this coefficient out before completing the square.
For example, solve by completing the square.
Since the leading coefficient is 2, factor this out to get .
We can now divide both sides by 2 to get .
Now completing the square we get .
Solving for 𝑥, we get .
Examples of Solving Quadratic Equations by Completing the Square
Here are some examples of solving quadratic equations by completing the square
How to Solve Quadratic Equations with the Quadratic Formula
The quadratic formula is the most reliable method for solving a quadratic equation. It can always be used to obtain solutions.
The quadratic formula
To use the quadratic formula, first find the values of b, b^{2}4ac and 2a so that these three values can be entered directly into the formula.
For example, solve the quadratic equation .
Here, , and .
Therefore , and .
Entering these values into the quadratic formula, .
Therefore and .
Solving Quadratic Equations with the Quadratic Formula Examples
Here are some examples of using the quadratic formula to solve quadratic equations:
Solving Quadratic Equations Not Written in Standard Form
To solve a quadratic equation that is not written in standard form, first rearrange it so that it is in the form a𝑥^{2}+b𝑥+c=0. Then it can be factorised or the quadratic formula can be used to solve it.
Solving a Quadratic Equation Not Written in Standard Form: Example 1
For example, solve the quadratic equation .
The first step is to multiply both sides of the equation by the denominator of .
The equation becomes .
Expanding the brackets, .
Subtracting 4 from both sides, .
can be factorised to give .
Therefore or .
Solving a Quadratic Equation Not Written in Standard Form: Example 2
Solve the quadratic equation, .
Expanding the brackets, .
Subtracting from both sides, .
We can solve this by finding the square root of both sides so that .
Solving a Quadratic Equation when given ‘y’
When given y in a quadratic equation, the corresponding values of x can be found by rearranging the equation equal to zero and solving for x.
For example, in the equation , find the values of for which .
Substituting into the equation, .
Adding 3 to both sides, .
Therefore, and .
How to Solve Quadratic Equations by Graphing
For a quadratic equation that is set equal to zero, the solutions are the xaxis intercepts of its graph. Graphing a quadratic is one way to find the xintercepts. These can be read off from the graph as the solutions to the quadratic equation.
For example, the graph of is shown below.
The graph has xintercepts of and .
These are the solutions of .
Solving Quadratic Equations with Complex Solutions
For a quadratic equation of the form a𝑥^{2}+bx+c=0, complex solutions will exist if b^{2}4ac<0. This is because it is not possible to obtain real answers when finding the square root of a negative number in the quadratic formula. Instead, the solutions will contain imaginary components.
For example, find the solutions to .
a=1, b=2 and c=5.
Therefore, the quadratic formula, becomes .
Since , .
Therefore simplifies to and so, .
Преобразование выражений, содержащих квадратный корень
Квадратный корень из произведения двух неотрицательных чисел, равен произведению квадратных корней из этих чисел.
Используя это свойство арифметического квадратного корня можно выносить изпод корня множитель.
Например: , так как , а
Существует операция внесение множителя под знак корня. Эта операция является обратной к операции вынесение множителя изпод знака корня. В данном случае мы осуществляем преобразование следующего вида:
Необходимо соблюдать условие, что a и b неотрицательные числа.
Например: , так как 4^{2} = 16
Освобождение от иррациональности в знаменателе
Суть метода состоит в умножении и делении дроби на такое выражение, которое позволит исключить иррациональность (квадратные и кубические корни) из знаменателя и сделает его проще. После этого дроби проще привести к общему знаменателю и окончательно упростить исходное выражение.
Алгоритм освобождения от иррациональности в знаменателе дроби:
 Разложить знаменатель дроби на множители.
 Если знаменатель имеет вид или содержит множитель , то числитель и знаменатель следует умножить на . Если знаменатель имеет вид или или содержит множитель такого вида, то числитель и знаменатель дроби следует умножить соответственно на или на . Числа и называют сопряжёнными.
 Преобразовать числитель и знаменатель дроби, если возможно, то сократить полученную дробь.
Как найти число, сопряжённое для числа . Попробуем это число представить в виде двучлена таким образом : — и выполнить преобразование:
Вопросы к конспектам
Найдите целое решение:
Избавиться от иррациональности в знаменателе дроби:
Избавиться от иррациональности в знаменателе дроби:
Последнее изменение: Воскресенье, 29 Январь 2017, 19:44
(fill in either box above to explore more about square roots)
Answers are rounded to seven decimal places.
 positive numbers: 2
 negative numbers: 3
 imaginary numbers: 7i
 complex numbers: 8+9i
Notes:
 The square root calculator provides the principal square root (the positive square root which is most commonly used).
Multiplying the principal square root by 1 will provide the negative square root if needed.  The square root of a negative number results in an imaginary number noted by the letter «i».
Square Root:
The square root of a number is a value that when multiplied by itself equals the original number.
Example: The square root of 9 = 3. Thus: 3 x 3 = 9.
Where A and B are valid numbers:
A is the square root of B if: A x A = B
Square Root Symbol / Square Root Sign:
The square root of a number N is commonly represented by:
It can also be represented by:
In exponential notation, it can be represented by: N^{1/2}
«Square root» can be abbreviated as «sqrt» Example: sqrt(2) = 1.4142136See: square root symbol if you need to add the symbol to a text document.
How are square roots used:
Square roots can be used:
 to solve for the distance between two points (Pythagorean Theorem)
 to solve for the length of a side of a right triangle (Pythagorean Theorem)
 to find the solutions to quadratic equations
 to find normal distribution
 to find standard deviation
 basically to solve for a squared variable in an equation
If you see an equation like N^{2} = 27 , you can solve for N by taking the square root of 27 which
gives N = 5.1961524
How to Calculate the Square Root of:
 A Positive Number:
The square root of a positive number can be calculated by: square root calculator above
 handheld calculator
 spreadsheet
 square roots without a calculator
 A Negative Number:
To calculate the square root of a negative number, find the square root of the same positive number and multiply by «i». ( where i represents an imaginary number and i = square root of 1)Example: square root of 5
= (square root of 5) x (square root of 1)
= (square root of 5) x (i)
= 2.236068 x i
= 2.236068i
 An Imaginary Number:
To calculate the square root of an imaginary number, find the square root of the number as if it were a real number (without the i) and then multiply by the square root of i (where the square root of i = 0.7071068 + 0.7071068i)Example: square root of 5i
= (square root of 5) x (square root of i)
= (2.236068) x (0.7071068 + 0.7071068i)
= 1.5811388 + 1.5811388i
 A Complex Number:
To find the square root of a complex number, use the following equation:Square root (a + bi) = c + di
Where:
c = (1/square root of 2) x square root of [ (square root of(a^{2}+b^{2})) + a ]
d = (sign of b/square root of 2) x square root of [ (square root of(a^{2}+b^{2})) — a ]Example: the square root of 35i = c + di
c = (1/square root of 2) x square root of [ (square root of(3^{2}+(5)^{2})) + 3 ]
c = 2.1013034
d = (1/square root of 2) x square root of [ (square root of(3^{2}+(5)^{2})) — 3 ]
d = 1.1897378
Thus: square root of 35i = 2.1013034 — 1.1897378i
Need to calculate the square root of a number with a handheld calculator or looking to buy a new calculator?
Most handheld calculators have a square root function.
Below is a review of some that are recommended and how to use them.
Graphing Calculator:
Texas Instruments TI84 Plus Graphing Calculator
To take the square root of a number, press [2ND] (the secondary function key) and then [] (the radical symbol key which is used to take the square root of a number) and then the number that you want to find the square root of and then the [ENTER] key.
(Note: This same method will also work with TI83 and TI81 calculators)
(To see what the graph looks like on this calculator,
click the show graph button below the calculator image on this page.)Pros:
It can be used on many college entrance exams (check your exam criteria).It is a popular calculator. (If you need help, the likelihood of finding someone who knows how to use it is higher).
The display shows seven lines of input/output. Long equations can be viewed and checked. (This is a nice benefit that graphing calculators have compared to scientific calculators that may only have a single line display.) Another benefit of having a large display is you can compare your current answer to past answers that are still on the screen. This can often help you discover an input error that may have otherwise gone unnoticed.
Cons:
It is bulkier than a scientific calculator.
It costs around $85 (USD) more than a scientific calculator.Price:
Best price for this calculator as of 922014 is around $94.00 (USD).Casio Graphing Calculator (fx9750GII)
To take the square root of a number, press [SHIFT] and then [] (the radical symbol is above the x^{2} key) and then the number that you want to find the square root of and then the [EXE] key.
(To see this graph, click the show graph button below the calculator image on this page.)
Pros:
The cost is half the price of a TI84 calculator.It is a little smaller than the TI84 calculator.
It can be used on many college entrance exams (check your exam criteria).
The display shows seven lines of input/output.
Cons:
It is not as popular as the TI84 calculator. (It may be harder to find someone to help if you have a question about how to use the calculator.)
Price:
Best price as of 922014 is around $42.74 (USD).
Scientific Calculator:
Texas Instruments TI36X Pro Scientific Calculator
To take the square root of a number, press [2ND] (the secondary function key) and then [] (the radical symbol key which is used to take the square root of a number) and then the number that you want to find the square root of and then the [ENTER] key.
Note: If your calculator is in «Math Print» mode, the display will show as the answer. To convert this answer to a decimal number, press the [<>~] key (the «Answer Toggle» key) which is the key above the [ENTER] key.Pros:
It is good if a graphing calculator is not permitted.
It is smaller and lighter than a graphing calculator.
The display shows four lines of input/output. Some scientific calculators only show one line of input/output.Cons:
The symbols shown on the function keys ( + , — , etc.) may be hard to see because they are the same silver color as the keys.Price:
Best price for this calculator on 922014 is around $19.99 (USD).
Math Help / Math Tutor
If you need help with math:
 There are a lot of online resources that are free to use and are available when you need them. Some sites have online courses, instructional videos, and even math games to help you learn more about math.
 Ask someone who is good at math to help. If you can’t find anyone who can help, a math teacher at a local school may be able to help or recommend a math tutor in your area.
 A local library may be able to help not only with informational books but they may be able to recommend a math tutor also.
 There are several websites that can help you find a local tutor if you would like inperson tutoring. Also, some tutors are able to help from a remote location if needed. Prices are typically between $10 to $50 (USD) per hour. Always research and try to find reviews, ratings, or feedback from others who have used a particular tutor before making a selection.