## 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 8^{3} 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:

- 5
^{4}= 5 × 5 × 5 × 5 = 625 - 3
^{5}= 3 × 3 × 3 × 3 × 3 = 243 - 14
^{2}= 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:

**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:

a^{m} × a^{n} = a^{m + n}, where m and n are real numbers

**Law of quotient**

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

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

**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.

**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

a^{0} = 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,

a^{n} b^{n} = (ab)^{n}

**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:

(a^{m })^{n} = a^{mn}

**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**^{3 }

** Solution:**

7^{3 }= 7 × 7 × 7 = 343

**Example 2: Express the result in exponential form.**

**2**^{3}** × 2**^{4}** × 2**^{7}** × 2**

**Solution:**

2^{3} × 2^{4} × 2^{7} × 2 = 2^{3 + 4 + 7 + 1} = 2^{15}

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) 2^{12} ÷ 2^{5}

(ii) 3^{3 }× 4^{3}

**Solution: **

(i) 2^{12} ÷ 2^{5 }= 2^{12 – 5} = 2^{7}= 128.

(ii) 3^{3 }× 4^{3 }= (3 × 4)^{ 3} = 1728

## Practice Problems

## Exponent## 1What is the value of (–3) |

## Frequently Asked Questions

**What are the real-life applications of exponent?**

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.

**What is the importance of exponent?**

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

**What are zero exponents?**

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

**When can exponents be added?**

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.

Read more about exponents and other interesting mathematical terms for children in grades Pre-k to 8 on SplashLearn.