# Exponent – Definition with Examples

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## Definition of Exponent

The exponent of a number indicates the total time to use that number in a multiplication. For example, 8 × 8 × 8  can be expressed as 83 because 8 is multiplied by itself 3 times. Here, 3 is the ‘exponent’ or ‘power’ which tells how many times 8 is multiplied by itself, and 8 is the ‘base’ which represents the number being multiplied. In short, power or exponent indicates the number of times a number needs to be multiplied by itself. Here, the base can be any integer, fraction or decimal. The exponent can also take up any value, be it positive or negative.

## Examples of Exponent

Here are some examples:

• 54 = 5 × 5 × 5 × 5 = 625
• 35 = 3 × 3 × 3 × 3 × 3 = 243
• 142 = 14 × 14 = 196
• (−4)3 = (−4) × (−4) × (−4) = −64
• $\left(\frac{1}{2} \right)^{4}$ = $\left(\frac{1}{2} \right)\times \left(\frac{1}{2} \right)\times \left(\frac{1}{2} \right)\times \left(\frac{1}{2} \right)$ =  $\frac{1}{16}$
• (0.2)3 = 0.2 × 0.2 × 0.2 = 0.008

## How to Read Numbers with Exponents

$x^{n}$ can be read in many ways.

• x to the power of n
• x raised to n
• x to the n
• x to the nth power

There are some special cases:

$x^{2}$ is also read as x square and $x^{3}$ is also read as x cube.

## Properties of Exponent

The rules or properties of exponents are used widely to solve various problems. Let us learn about them one by one:

1. Law of product

As per the law, to find the product of exponential expressions with the same base we add the exponents. It is given as:

am × an = am + n, where m and n are real numbers

1. Law of quotient

As per the law, to divide two exponential expression with the same base we subtract the exponents. It is given as:

$\frac{a^{m}}{a^{n}}$ = am – n, where m and n are real numbers and a is a non-zero term.

1. Law of negative exponent

The negative exponent rule states that when an exponent is negative, we can convert it into positive by reciprocating it. It is represented as:

a–m = $\frac{1}{a^{m}}$, where a is a non-zero number and m is a real number.

1. Law of zero exponent

As per this law, if the exponent of a real number is 0, then its value is equal to 1, as

a0 = 1.

1. Law of power of a product

As per this law, multiplying two different bases with the same power is equal to the product of the base with the power. For instance,

an bn = (ab)n

1. Law of power of a power

As per this law, when the power raised to the another power of a base we multiply the exponents. It is represented as:

(am )n = amn

1. Law of power of a quotient

According to this law, the division of different bases with the same power is represented as

a$\frac{a^{n}}{b^{n}}$ = $\left (\frac{a}{b}\right)^n$, where b is non-zero term.

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## Fun Facts

• A base with 0 as its power is always equal to 1.
• Any number raised to power 1 is always equal to that number.
• Any power raised to the base 0 is always equal to 0.
• The history of power or exponent is quite old. The term was used for the first time in 1544.

## Solved Examples

Example 1: Evaluate 7

Solution:

73 = 7 × 7 × 7 = 343

Example 2: Express the result in exponential form.

23 × 24 × 27 × 2

Solution:

23 × 24 × 27 × 2 = 23 + 4 + 7 + 1 = 215

Example 3: Simplify $\Bigg\{\Big(\frac{2}{3}\Big)^{3}\Bigg\}^{-2}$

Solution:

$\Bigg\{\Big(\frac{2}{3}\Big)^{3}\Bigg\}^{-2}$ = $\frac{2}{3}^{-6}$ = $\frac{3}{2}^{6}$ = $\frac{729}{64}$

Example 4: Simplify the following

(i) 212 ÷ 25

(ii) 33 × 43

Solution:

(i) 212 ÷ 25 = 212 – 5 = 27= 128.

(ii) 33 × 43 = (3 × 4) 3 = 1728

## Practice Problems

Exponents have various applications. A few of them are mentioned below:

• Scientific scales like the Richter scale and Ph scale are based on exponents.
• They are widely used to calculate volume, area, and measurement-related problems.
• They are also used in computer games.

Exponents are important to write the values of numbers in simplified form. Repeated multiplication can be simply written with the help of exponents.

Zero exponents are the numbers with 0 as their exponent. Any base with 0 as its exponent is equal to 1.

When exponents with the same base are multiplied, we can add their powers.

## Conclusion

In this article, we have learnt about exponents and their various laws. The exponent of a number indicates the total time to use that number in a multiplication. They are important to write the values of numbers in simplified form.

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