Exponent Rules: Negative Exponents

Overview

Negative exponents indicate that the base should be moved to the opposite part of the fraction (numerator to denominator or vice versa), and the exponent becomes positive. In mathematical terms:

\( a^{-n} = \frac{1}{a^n} \) (for \( a \neq 0 \))

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

This rule applies to both numbers and variables with negative exponents.

Steps to Simplify Expressions with Negative Exponents

  1. If the base with the negative exponent is in the numerator, move it to the denominator and make the exponent positive.
  2. If the base with the negative exponent is in the denominator, move it to the numerator and make the exponent positive.
  3. Simplify the remaining expression.

Example 1: Simplifying with a Single Term

Consider the following expression:

\( 2^{-3} \)

Step 1: Move the base to the denominator and make the exponent positive:

\( \frac{1}{2^3} \)

Step 2: Simplify:

\( \frac{1}{8} \)

Example 2: Simplifying with Variables

Consider the following expression:

\( x^{-4} \)

Step 1: Move the base to the denominator and make the exponent positive:

\( \frac{1}{x^4} \)

Example 3: Simplifying with Fractions

Consider the following expression:

\( \frac{1}{y^{-2}} \)

Step 1: Move the base to the numerator and make the exponent positive:

\( y^2 \)

Practice Questions

  1. Question 1: Simplify the following expression:

    \( 5^{-2} \)

    Solution

    Step 1: Move the base to the denominator and make the exponent positive:

    \( \frac{1}{5^2} \)

    Step 2: Simplify:

    \( \frac{1}{25} \)

  2. Question 2: Simplify the following expression:

    \( a^{-3} \)

    Solution

    Step 1: Move the base to the denominator and make the exponent positive:

    \( \frac{1}{a^3} \)

  3. Question 3: Simplify the following expression:

    \( \frac{1}{b^{-4}} \)

    Solution

    Step 1: Move the base to the numerator and make the exponent positive:

    \( b^4 \)

  4. Question 4: Simplify the following expression:

    \( (2x)^{-2} \)

    Solution

    Step 1: Move the base to the denominator and make the exponent positive:

    \( \frac{1}{(2x)^2} \)

    Step 2: Simplify:

    \( \frac{1}{4x^2} \)

  5. Question 5: Simplify the following expression:

    \( (3y^2)^{-1} \)

    Solution

    Step 1: Move the base to the denominator and make the exponent positive:

    \( \frac{1}{3y^2} \)

→Fractional Exponents