Understanding  Negative Exponents

A negative exponent indicates that we take the reciprocal of the base and then make the exponent positive. In other words, for any nonzero number a, $a^{-n} = 1/ a^n$. They are commonly used in algebra and for scientific calculations. Understanding negative exponents makes it easier to simplify expressions and solve problems involving negative powers. In this guide, you will learn the definition of negative exponents, their rules, and step-by-step methods to solve problems with clear examples.

Table of Contents

• What are Negative Exponents
• Solved Examples on Negative Exponents
• Practice Questions on Negative Exponent
• Frequently Asked Questions on Negative Exponents

What are Negative Exponents

Study the following chart:

We know the meaning of exponents with non-negative integral powers.

But what does 7⁻¹ or 7⁻⁴ mean? That is, when the value of the power is a negative number.

We can find the values of 7⁻¹ and 7⁻⁴ by extending the above chart.

Hence, 7⁻¹ =  $1/7$  and    7⁻⁴ = $1/7×7×7×7$

We can write the denominators of these numbers in exponential form as:

7⁻¹ =  $1/7^1$  and    7⁻⁴ =  $1/7^4$

Similarly,   7⁻² =  $1/7^2$ and    7⁻³ =  $1/7^2$

In general, for any non-zero number x,   x^−m = 1/x^m   also,   x^−m × x^m = 1

Important Note: When the base is a rational number   a/b,   (a/b)⁻ⁿ = (b/a)ⁿ

Solved Examples on Negative Exponents

Example 1: Find the multiplicative inverse of 11⁷ .
Solution: x^m and x^−m are the multiplicative inverses of each other.
The multiplicative inverse of 11⁷ is 11⁻⁷.
Example 2: Find the multiplicative inverse of 2⁻⁵ .
Solution: x^m and x^−m are the multiplicative inverses of each other.
The multiplicative inverse of 2⁻⁵ is 2⁵.
Example 3: Find the multiplicative inverse of 27⁻¹⁴.
Solution: x^m and x^−m are the multiplicative inverses of each other.
The multiplicative inverse of 27^-14 is 27¹⁴.

Practice Questions on Negative Exponent

1.   Find the multiplicative inverse of   $6^{12}$
2. Find the multiplicative inverse of  $22^{-3}$
3. Find the multiplicative inverse of  $52^6$