Maths Commands in LaTeX

Maths Commands in LaTeX Техника

Learning Outcomes

  • Simplify Nth roots.
  • Write radicals as rational exponents.

Maths Commands in LaTeX

To write in latex in mathematical language you need a base. In this
section you will find the basic commands to write the main operations. Below there is a
list of the topics we will touch.

fractions

exponentiations


subscripts


roots

logarithms


summations and products

limits


derivatives


integrals

1) Write the steps to find the hypotenuse length of a 4cm base right triangle and 3cm high by using
the Pitagora Theorem.
Then, in a second line, multiply it by the logarithm of base 10 of 100.

2) Using a summation, write the sum of the first 5 numbers raised to the power of 3/2, starting with 1.

author

  • Constructions in LaTeX are used to construct various parts of an equation such as fraction, vector, integration, square root, etc. Use of LaTeX codes is mandatory to write them.

    Some constructions along with their LaTeX codes :

    Greek Letters in LaTeX

    Maths Commands in LaTeX

    Geometry Notations in LaTeX

    Set Notations in LaTeX

    Brackets in Latex

    Logic Notations in LaTeX

    Relational Operators in LaTeX

    Trigonometric Functions in Latex

    Arrow Symbols in LaTeX

    Fonts in LaTeX

    Miscellaneous symbols in LaTeX

    Probabilistic Version Of Random-Walk

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    A General Note: Principal nth Root

    Example 10: Simplifying nth Roots

    1. [latex]\sqrt[5]{-32}[/latex]
    2. [latex]\sqrt[4]{4}\cdot \sqrt[4]{1,024}[/latex]
    3. [latex]-\sqrt[3]{\frac{8{x}^{6}}{125}}[/latex]
    4. [latex]8\sqrt[4]{3}-\sqrt[4]{48}[/latex]
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    Solution

    1. [latex]\sqrt[5]{-32}=-2[/latex] because [latex]{\left(-2\right)}^{5}=-32 \\ \text{ }[/latex]
    2. First, express the product as a single radical expression. [latex]\sqrt[4]{4,096}=8[/latex] because [latex]{8}^{4}=4,096 \\[/latex]
    3. [latex]\begin{array}{cc}\\ \frac{-\sqrt[3]{8{x}^{6}}}{\sqrt[3]{125}}\hfill & \text{Write as quotient of two radical expressions}.\hfill \\ \frac{-2{x}^{2}}{5}\hfill & \text{Simplify}.\hfill \\ \end{array}[/latex]
    4. [latex]\begin{array}{cc}\\ 8\sqrt[4]{3}-2\sqrt[4]{3}\hfill & \text{Simplify to get equal radicands}.\hfill \\ 6\sqrt[4]{3} \hfill & \text{Add}.\hfill \\ \end{array}[/latex]

    Try It 10

    1. [latex]\sqrt[3]{-216}[/latex]
    2. [latex]\frac{3\sqrt[4]{80}}{\sqrt[4]{5}}[/latex]
    3. [latex]6\sqrt[3]{9,000}+7\sqrt[3]{576}[/latex]

    Learning Outcomes

    • Simplify Nth roots.
    • Write radicals as rational exponents.

    Recall: operations on Fractions

    When simplifying handling nth roots and rational exponents, we often need to perform operations on fractions. It’s important to be able to do these operations on the fractions without converting them to decimals. Recall the rules for operations on fractions.

    • To multiply fractions, multiply the numerators and place them over the product of the denominators.
      •  [latex]\dfrac{a}{b}\cdot\dfrac{c}{d} = \dfrac {ac}{bd}[/latex]
    • To divide fractions, multiply the first by the reciprocal of the second.
      •  [latex]\dfrac{a}{b}\div\dfrac{c}{d}=\dfrac{a}{b}\cdot\dfrac{d}{c}=\dfrac{ad}{bc}[/latex]
    • To simplify fractions, find common factors in the numerator and denominator that cancel.
      •  [latex]\dfrac{24}{32}=\dfrac{2\cdot2\cdot2\cdot3}{2\cdot2\cdot2\cdot2\cdot2}=\dfrac{3}{2\cdot2}=\dfrac{3}{4}[/latex]
    • To add or subtract fractions, first rewrite each fraction as an equivalent fraction such that each has a common denominator, then add or subtract the numerators and place the result over the common denominator.
      •  [latex]\dfrac{a}{b}\pm\dfrac{c}{d} = \dfrac{ad \pm bc}{bd}[/latex]
  • LATEX is a document preparation system for producing professional-looking documents. LaTeX is widely used for the communication and publication of scientific documents in many fields, including mathematics, statistics, computer science, engineering, physics, etc. It also has a prominent role in the preparation and publication of books and articles that contain complex multilingual materials, such as Sanskrit and Greek.

    So in this post we have discussed the most used TEX commands used for Maths.

    1. Fractions:
      Instead of writing fractions as A / B we will use below syntax
      Syntax :
      \frac{numerator}{denominator}
      
      \frac{a+1}{b+1}
      
    2. Nth power:
      Instead of writing powers as x ^ n which is not clear as if it is xor or power so we will use below syntax
      Syntax:
      x^y
      
      x^2
      
    3. Nth root:
      Instead of writing roots as x^(1/N) which is not clear as if it is xor or root so we will use below syntax
      Syntax:
      \sqrt[N]{27}
      
      \sqrt[3]{27}
      
    4. Matrices
      Instead of writing matrices as [[1, x, x^2], [1, y, y^2][1, z, z^2]] which is not very clear use below syntax
      Syntax:
      \begin{matrix}
          1 & x & x^2 \\
          1 & y & y^2 \\
          1 & z & z^2 \\
      \end{matrix}
      
      \begin{matrix}
          1 & x & x^2 \\
          1 & y & y^2 \\
          1 & z & z^2 \\
      \end{matrix}
      
    5. Definitions by cases (piecewise function) is a function defined by multiple sub-functions, each sub-function applying to a certain interval of the main function’s domain, a sub-domain.
       f(n) =
      \begin{cases}
      n/2,  & \text{if $n$ is even} \\
      n+1, & \text{if $n$ is odd}
      \end{cases}
      
       f(n) =
      \begin{cases}
      n/2,  & \text{if $n$ is even} \\
      n+1, & \text{if $n$ is odd}
      \end{cases}
      
    6. System of equations is a function defined by multiple sub-functions, each sub-function applying to a certain interval of the main function’s domain, a sub-domain.
       \left\{ 
      \begin{array}{c}
      a_1x+b_1y+c_1z=d_1 \\ 
      a_2x+b_2y+c_2z=d_2 \\ 
      a_3x+b_3y+c_3z=d_3
      \end{array}
      \right. 
      
       \left\{ 
      \begin{array}{c}
      a_1x+b_1y+c_1z=d_1 \\ 
      a_2x+b_2y+c_2z=d_2 \\ 
      a_3x+b_3y+c_3z=d_3
      \end{array}
      \right. 
      
    7. Summation is the addition of a sequence of any kind of numbers, called addends or summands; the result is their sum or total.
      \sum_{i=0}^n i^2
      
      \sum_{i=0}^n i^2
      
    8. subscriptsis a character that is set slightly below the normal line of type.
      \log_2 x
      
      \log_2 x
      
    9. floor is the function that takes as input a real number and gives as output the greatest integer less than or equal to, denoted.
      \lfloor n \rfloor
      
      \lfloor 2.2 \rfloor
      
    10. ceil function maps to the least integer greater than or equal to, denoted.
      \lceil n \rcei
      
      \lceil 2.5 \rceil
      
    11. Some Combined examples :
        Example –

      • Use
        \sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}
      • Use
        \left(\frac{\sqrt x}{y^3}\right)
      • Use
        \Biggl(\biggl(\Bigl(\bigl((n)\bigr)\Bigr)\biggr)\Biggr)
      • Use
        \sqrt[3]{\frac xy}

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    Using Rational Roots

    A General Note: Principal nth Root

    Example: Simplifying nth Roots

    1. [latex]\sqrt[5]{-32}[/latex]
    2. [latex]\sqrt[4]{4}\cdot \sqrt[4]{1,024}[/latex]
    3. [latex]-\sqrt[3]{\dfrac{8{x}^{6}}{125}}[/latex]
    4. [latex]8\sqrt[4]{3}-\sqrt[4]{48}[/latex]

    Using Rational Roots

    A General Note: Principal nth Root

    Example: Simplifying nth Roots

    1. [latex]\sqrt[5]{-32}[/latex]
    2. [latex]\sqrt[4]{4}\cdot \sqrt[4]{1,024}[/latex]
    3. [latex]-\sqrt[3]{\dfrac{8{x}^{6}}{125}}[/latex]
    4. [latex]8\sqrt[4]{3}-\sqrt[4]{48}[/latex]

    Try It

    1. [latex]\sqrt[3]{-216}[/latex]
    2. [latex]\dfrac{3\sqrt[4]{80}}{\sqrt[4]{5}}[/latex]
    3. [latex]6\sqrt[3]{9,000}+7\sqrt[3]{576}[/latex]

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    Using Rational Exponents

    We can also have rational exponents with numerators other than 1. In these cases, the exponent must be a fraction in lowest terms. We raise the base to a power and take an nth root. The numerator tells us the power and the denominator tells us the root.

    All of the properties of exponents that we learned for integer exponents also hold for rational exponents.

    Rational Exponents

    Rational exponents are another way to express principal nth roots. The general form for converting between a radical expression with a radical symbol and one with a rational exponent is

    How To: Given an expression with a rational exponent, write the expression as a radical.

    1. Determine the power by looking at the numerator of the exponent.
    2. Determine the root by looking at the denominator of the exponent.
    3. Using the base as the radicand, raise the radicand to the power and use the root as the index.

    Example: Writing Rational Exponents as Radicals

    Try It

    Example: Writing Radicals as Rational Exponents

    Try It

    Watch this video to see more examples of how to write a radical with a fractional exponent.

    https://youtube.com/watch?v=L5Z_3RrrVjA%3Ffeature%3Doembed

    Example: Simplifying Rational Exponents

    1. [latex]5\left(2{x}^{\frac{3}{4}}\right)\left(3{x}^{\frac{1}{5}}\right)[/latex]
    2. [latex]{\left(\dfrac{16}{9}\right)}^{-\frac{1}{2}}[/latex]

    Try It

    Using Rational Exponents

    We can also have rational exponents with numerators other than 1. In these cases, the exponent must be a fraction in lowest terms. We raise the base to a power and take an nth root. The numerator tells us the power and the denominator tells us the root.

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    All of the properties of exponents that we learned for integer exponents also hold for rational exponents.

    Rational Exponents

    Rational exponents are another way to express principal nth roots. The general form for converting between a radical expression with a radical symbol and one with a rational exponent is

    How To: Given an expression with a rational exponent, write the expression as a radical.

    1. Determine the power by looking at the numerator of the exponent.
    2. Determine the root by looking at the denominator of the exponent.
    3. Using the base as the radicand, raise the radicand to the power and use the root as the index.

    Example: Writing Rational Exponents as Radicals

    Example: Writing Radicals as Rational Exponents

    Try It

    Watch this video to see more examples of how to write a radical with a fractional exponent.

    https://youtube.com/watch?v=L5Z_3RrrVjA%3Ffeature%3Doembed%26rel%3D0

    Example: Simplifying Rational Exponents

    1. [latex]5\left(2{x}^{\frac{3}{4}}\right)\left(3{x}^{\frac{1}{5}}\right)[/latex]
    2. [latex]{\left(\dfrac{16}{9}\right)}^{-\frac{1}{2}}[/latex]

    Try It

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