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.