# Power in Math || Basic rules of power and exponents

In this post we will understand the concept of power in math and its various properties.

## What is Power in Math?

When a number is multiplied by itself we get power.

For Example
\mathtt{2\times 2\times 2\times 2\times 2\ =\ 2^{5}}

Here number 2 is multiplied by itself 5 times.
This can be simply written as \mathtt{\ 2^{5}}

The number which is multiplied is called Base.
Here number 2 is the base

The number of times a base is multiplied is called Power
Here number 5 is power.

Note:
The numbers with power is also called exponents

Given below are some examples of power used in algebra.

(a) \mathtt{3\times 3\times 3\times 3\times 3\times 3\ =\ 3^{6}}

(b) \mathtt{y\times y\times y\times y\ =\ y^{4}}

(c) \mathtt{m\times m\times n\times n\times n\ =\ m^{2} n^{3}}

(d) \mathtt{-6\times p\times p\times q\times r\times r\ =\ -6p^{2} qr^{2}}

## Important rules for Power in algebra

Given below are some of the rules for power which would be helpful to solve algebraic equation.

You have to remember each of the given rules for your examination.

(1) Multiplication of same base

Multiplication of numbers with different power can be easily done by adding the powers.

Example 01
\mathtt{( 3)^{4} \times ( 3)^{2} \ }

Solution
\mathtt{\Longrightarrow ( 3)^{4\ +\ 2}}\\\ \\ \mathtt{\Longrightarrow \ ( 3)^{6}}

Example 02
\mathtt{( 5)^{2} \times ( 5)^{3} \times ( 5)^{6} \ }

Solution
\mathtt{\Longrightarrow ( 5)^{2+3+\ 6}}\\\ \\ \mathtt{\Longrightarrow \ ( 5)^{11} \ }

(02) Division of Exponents

The division of numbers with power can be done by simply subtracting the powers.

Example 01
\mathtt{( 6)^{4} \div ( 6)^{3} \ }

Solution
\mathtt{\Longrightarrow ( 6)^{4-3}}\\\ \\ \mathtt{\Longrightarrow \ ( 6)^{1} \ }

Example 02
\mathtt{( 9)^{3} \div ( 9)^{5} \ }

Solution
\mathtt{\Longrightarrow ( 9)^{3-5}}\\\ \\ \mathtt{\Longrightarrow \ ( 9)^{-2} \ }

(03) Power of Power

When number with power is raised to another power, the simplification can be done by multiplying both the powers.

Example 01
Simplify the exponent \mathtt{\left( 2^{3}\right)^{5}}

Solution
\mathtt{\Longrightarrow ( 2)^{3\times 5}}\\\ \\ \mathtt{\Longrightarrow \ ( 2)^{15}}

Example 02
Simplify \mathtt{\left( 11^{4}\right)^{9} \ }

Solution
\mathtt{\Longrightarrow ( 11)^{4\times 9}}\\\ \\ \mathtt{\Longrightarrow \ ( 11)^{36} \ }

(04) Multiplication of number with same power

The multiplication of numbers with same power can easily be done by multiplying the numbers and leaving the power as it is.

Example 01
\mathtt{( 5)^{2} \times ( 8)^{2} \ }

Solution
\mathtt{\Longrightarrow ( 8\times 5)^{2}}\\\ \\ \mathtt{\Longrightarrow \ ( 40)^{2} \ }

Example 02
\mathtt{( 3)^{3} \times ( 4)^{3} \times ( 5)^{3} \ }

Solution
\mathtt{\Longrightarrow ( 3\times 4\times 5)^{3}}\\ \\ \mathtt{\Longrightarrow \ ( 60)^{3} \ }

(05) Negative Power

The negative power of number can be converted into positive power by taking the reciprocal of given number or vice-versa

Example 01
Convert the negative power into positive one .
\mathtt{\Longrightarrow ( 2)^{-6}}

Solution
\mathtt{\Longrightarrow \frac{1}{2^{6}}}

Example 02
Simplify the negative power
\mathtt{\Longrightarrow \frac{1}{3^{-5}}}

Solution
\mathtt{\Longrightarrow \ 3^{5}}

(06) Power of 1

Any number raised to the power of 1 results in same number.

Examples
\mathtt{9^{1} \Longrightarrow \ 9}\\\ \\ \mathtt{13^{1} \Longrightarrow \ 13}\\\ \\ \mathtt{17^{1} \Longrightarrow \ 17}

(07) Power of 0

Any number raised to the power 0 results in number 1.

Examples
\mathtt{10^{0} \Longrightarrow \ 1}\\\ \\ \mathtt{-3^{0} \Longrightarrow \ -1}\\\ \\ \mathtt{7^{0} \Longrightarrow \ 1}

(08) Fractional exponents

If any number is raised to the power of fractional number then:

(a) The denominator of the power represent the root of number.
If denominator is 2, then the power can be converted into square root.
If denominator is 3, then power is converted into cube root and vice-versa.

(b) the numerator of fraction simply represents the power.

Example 01
Simplify \mathtt{9^{\frac{3}{2}}}

Solution
\mathtt{\Longrightarrow \left( \ \sqrt[2]{9}\right)^{3}}\\\ \\ \mathtt{\Longrightarrow \ ( 3)^{3}}\\\ \\ \mathtt{\Longrightarrow \ 27}

Example 02
\mathtt{16^{\frac{5}{4}}}

Solution
\mathtt{\Longrightarrow \left( \ \sqrt[4]{16}\right)^{5}}\\\ \\ \mathtt{\Longrightarrow \ ( 2)^{5}}\\\ \\ \mathtt{\Longrightarrow \ 32}

## Frequently asked Questions – Exponents

(01) What is zero exponent?

Any number raised to the power 0 is zero exponent.

The value of number with power 0 is 1.

(02) Can we have numbers with decimal powers?

Yes!!
Given below are some of the examples;

\mathtt{\Longrightarrow 3^{2.5}}\\\ \\ \mathtt{\Longrightarrow 9^{1.6}}\\\ \\ \mathtt{\Longrightarrow 7^{2.3}}

Can we simplify the decimal power?

Yes!!
The decimal power can be simplified by converting into fraction.

Example
Simplify \mathtt{16^{1.5}}

Solution
Convert 1.5 into fraction

\mathtt{\Longrightarrow 16^{\frac{15}{10}}}

Dividing numerator and denominator by 5

\mathtt{\Longrightarrow 16^{\frac{3}{2}}}\\\ \\ \mathtt{\Longrightarrow \left(\sqrt[2]{16} \ \right)^{3}}\\\ \\ \mathtt{\Longrightarrow \ \ ( 4)^{3}}\\\ \\ \mathtt{\Longrightarrow \ 64\ }

(03) How are exponents useful in our daily life?

Writing repeated multiplication of numbers can be very tiring and problematic.

Using the concept of exponents help us to compress long multiplication into single digit.

(04) What is one exponent?

Number raised to the power 1 is one exponent.

Any number with power 1 results in same number.