In this chapter we will learn about the **concept of exponents** with solved examples.

## Exponents – What are they ?

Exponent is the **way to represent repeated multiplication of number by itself.**

The exponent is also called** Power**. So don’t get confused with the above two terms.

Suppose we multiply number 3 by itself 5 times.

\mathtt{\Longrightarrow \ 3\times 3\times 3\times 3\times 3}

Using exponents, this expression can be written as;

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

Hence, **exponent is the short hand notation of multiplication expression in which numbers are multiplied by itself**.

### Base & Power in exponent function

In the exponent function, **you will see large number in bottom and small number in the upper right corner**.

The large number is called the **Base**.

The small number at the top is called** Power / Exponent**.**For example;**

Consider the exponent function \mathtt{7^{3}}

Here number 7 is the base and number 3 is the exponent.

### How to read exponents in words ?

There are** many ways to read an exponent function**.

Consider the expression \mathtt{9^{5}} .

You can read the exponent in following ways;

(i) “9 to the power 5”

(ii) ” 9 raised to the 5 “

(iii) ” 9 to the 5th “

(iv) ” 9 to the 5 “

Hence, there are multiple ways to express the exponent number. So don’t get confused with different sentences, they all mean the same thing.

## How to simplify exponents ?

Just look at the base and power of given exponent function.

Here, **base tells the number that is multiplied** and **power tells the number of times it is multiplied**.

Let us understand this with examples;**Example 01**

Exponent \mathtt{6^{5}}

**Solution**

Base = 6

It means number 6 is multiplied repeatedly.

Power = 5

It tells that the multiplication is done 5 times.

Simplifying the above exponents;

\mathtt{\Longrightarrow \ 6^{5}}\\\ \\ \mathtt{\Longrightarrow \ 6\times 6\times 6\times 6\times 6}\\\ \\ \mathtt{\Longrightarrow \ 7776}

Hence, **7776 is the solution of given exponents.**

**Example 02**

Simplify exponent \mathtt{11^{3}}

**Solution**

Base is 11.

Here number 11 is multiplied repeatedly.

Exponent is 3.

It means that the number is multiplied thrice.

Simplifying the exponents;

\mathtt{\Longrightarrow \ 11^{3}}\\\ \\ \mathtt{\Longrightarrow \ 11\times 11\times 11}\\\ \\ \mathtt{\Longrightarrow \ 1331}

Hence, **1331 is the solution of given exponents.**

### Exponent function with negative base

There are two possibilities for exponents with negative base;

(a) Negative base with **even power**

In this case the final result is a positive number.

(b) Negative base with** odd power**

In this case the final result is negative number.

Both the above cases are concluded in below diagram.

Let us see some examples for further understanding.**Example 01**

Find the value of exponent \mathtt{\ ( -2)^{3}}

**Solution**

Here the number has negative base and odd power. So the final result will be a negative number.

\mathtt{\Longrightarrow \ ( -2)^{3}}\\\ \\ \mathtt{\Longrightarrow \ -2\times -2\times -2}\\\ \\ \mathtt{\Longrightarrow \ -8}

Hence, **-8 is the final solution**.

**Example 02**

Find the value of exponent \mathtt{( -5)^{4}} **Solution**

Here the number has negative base with even power. So the final result will be a positive number.

\mathtt{\Longrightarrow \ ( -5)^{4}}\\\ \\ \mathtt{\Longrightarrow \ -5\times -5\times -5\times -5}\\\ \\ \mathtt{\Longrightarrow \ 625}

Hence, **625 is the final solution**.

### Exponent function with negative power

The exponent with **negative power can be converted into positive one by taking the reciprocal**.

Let \mathtt{\ ( a)^{-n}} be the given exponent.

Then we can write;

\mathtt{\ ( a)^{-n} \ =\ \frac{1}{a^{n}}}

Let us see some examples related to this concept;**Example 01**

Solve the exponent \mathtt{( 2)^{-3}} .**Solution**

First convert the negative exponent into positive one by taking reciprocal.

\mathtt{\Longrightarrow \ ( 2)^{-3} \ }\\\ \\ \mathtt{\Longrightarrow \ \ \frac{1}{2^{3}}}

Now expand the exponent function.

\mathtt{\Longrightarrow \ \ \frac{1}{2^{3}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{2\times 2\times 2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{8}}

Hence, **1/8 is the solution.**

**Example 02**

Solve the exponent \mathtt{( 11)^{-2}} **Solution**

Convert the power from negative into positive by taking reciprocal.

\mathtt{\Longrightarrow \ ( 11)^{-2} \ }\\\ \\ \mathtt{\Longrightarrow \ \ \frac{1}{11^{2}}}

Now calculate the value of exponent.

\mathtt{\Longrightarrow \ \ \frac{1}{11^{2}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{11\times 11}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{121}}

Hence, **1/121 is the solution.**

I hope you understood all the above concepts. Let us now solve some problems.

## Exponents – Solved Problems

**(01) Find the value of below exponents.**

\mathtt{( i) \ ( 2)^{5}}\\\ \\ \mathtt{( ii) \ ( -3)^{3}}\\\ \\ \mathtt{( iii) \ ( -5)^{2}}\\\ \\ \mathtt{( iv) \ ( 7)^{-4}}\\\ \\ \mathtt{( v) \ ( 10)^{0}}

**Solution**

\mathtt{( i) \ \ ( 2)^{5}}\\\ \\ \mathtt{\Longrightarrow 2\times 2\times 2\times 2\times 2\ }\\\ \\ \mathtt{\Longrightarrow \ 32}

\mathtt{( ii) \ ( -3)^{3}}\\\ \\ \mathtt{\Longrightarrow -3\times -3\times -3\ }\\\ \\ \mathtt{\Longrightarrow \ -27}

\mathtt{( iii) \ ( -5)^{2}}\\\ \\ \mathtt{\Longrightarrow -5\times -5}\\\ \\ \mathtt{\Longrightarrow \ 25}

\mathtt{( iv) \ ( 7)^{-4}}\\\ \\ \mathtt{\Longrightarrow \frac{1}{7^{4}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{7\mathtt{\times 7\times 7\times } 7}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{2401}}

\mathtt{( v) \ ( 10)^{0}}

Any number with power 0 results in number 1.

\mathtt{( 10)^{0} \ \Longrightarrow \ 1}

**(02) Express the below numbers in exponent form**

(i) -125

(ii) 64

(iii) 14641

(iv) 4/81

(v) -100/ 1728**Solution**** (i) -125 **

\mathtt{\Longrightarrow \ -125}\\\ \\ \mathtt{\Longrightarrow \ -5\times -5\times -5}\\\ \\ \mathtt{\Longrightarrow \ ( -5)^{3}}

**(ii) 64 **

\mathtt{\Longrightarrow \ 64}\\\ \\ \mathtt{\Longrightarrow \ 8\times 8}\\\ \\ \mathtt{\Longrightarrow \ ( 8)^{2}}

**(iii) 14641**

\mathtt{\Longrightarrow \ 14641}\\\ \\ \mathtt{\Longrightarrow \ 11\times 11\times 11\times 11}\\\ \\ \mathtt{\Longrightarrow \ ( 11)^{4}}

** (iv) 4/81 **

\mathtt{\Longrightarrow \ \frac{4}{81}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2\times 2}{9\times 9}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2^{2}}{9^{2}}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{2}{9}\right)^{2}}

**(v) -100/ 1728 **

\mathtt{\Longrightarrow \ \frac{-100}{1728}}\\\ \\ \mathtt{\Longrightarrow \ \frac{10\times 10}{-12\times -12\times -12}}\\\ \\ \mathtt{\Longrightarrow \ \frac{10^{2}}{-12^{3}}}