Negative Exponents

A negative exponent is an exponent with a negative value. It indicates that the base should be written as its reciprocal with a positive exponent.

Example: Simplify

1. 2^-3
2. (2/3)^-3

Solution:

Example 1:

3^{-2} = \frac{1}{3^2} = \frac{1}{(3 \times 3)}

= \frac{1}{9} = 0.111\ldots

Example 2:

\left(\frac{3}{5}\right)^{-2} = \left(\frac{5}{3}\right)^2

= \left(\frac{5}{3}\right)^2= \frac{5^2}{3^2} = \frac{25}{9} \approx 2.78

Negative Exponent Rules

There are two basic rules for simplifying negative exponents.

Rule 1: Negative Exponent in the Numerator

A negative exponent can be converted into a positive exponent by taking the reciprocal of the base.

a^(-n) = 1/a × 1/a × ... n times = 1/aⁿ

Rule 2: Negative Exponent in the Denominator

A negative exponent in the denominator can be moved to the numerator by changing its exponent to positive.

1/a^(-n) = a × a × ... .n times = aⁿ

The negative exponent's rule can be easily understood with the example discussed in the image below,

These rules can be easily understood by the example discussed below,

Example: Simplify 3^-3 × 1/(4^-2)

Solution:

Using the above rule for solving negative exponents,

a^-n = 1/aⁿ and 1/a^(-n) = aⁿ

3^-3 = 1/3³ 

     = 1/27

1/(4^-2) = 4²

          = 16

Now, 3^-3 × 1/(4^-2)

= 1/27 × 16

= 16/27

Negative Fraction Exponents

A negative fractional exponent represents the reciprocal of a root.

Simplify: 125^{-\frac{1}{3}}

Solution:

125^{-\frac{1}{3}}

= \frac{1}{\sqrt[3]{125}} = \frac{1}{5}

Multiplying Negative Exponents

Negative exponents follow the same multiplication and division rules as positive exponents.

• Multiplication Rule: When multiplying powers with the same base, add their exponents.
• Division Rule: When dividing powers with the same base, subtract the denominator's exponent from the numerator's exponent.

Simplify:

1. 2⁻³ × 2⁻²
2. 3⁻² ÷ 3⁻⁴

Solution:

Example 1: (Formula used: aᵐ × aⁿ = aᵐ⁺ⁿ)

2^{-3 + (-2)}
= 2⁻⁵
= 1/2⁵
= 1/32

Example 2: (Formula used: aᵐ ÷ aⁿ = aᵐ⁻ⁿ)

3^-2-(-4)
= 3²
= 9

Practice Questions on Negative Exponents