Exponent Rules: Power of a Quotient

Overview

The Power of a Quotient rule states that when raising a fraction (quotient) to a power, both the numerator and denominator are raised to the same power. In mathematical terms:

\(\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}\)

This rule applies when a fraction inside parentheses is raised to an exponent.

Steps to Apply the Power of a Quotient Rule

  1. Distribute the exponent to both the numerator and denominator of the fraction.
  2. Simplify each term if possible.

Example 1: Simplifying with Numbers

Consider the following expression:

\(\left( \frac{3}{2} \right)^2\)

Step 1: Distribute the exponent 2 to both the numerator and denominator:

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

Step 2: Simplify:

\(\frac{9}{4}\)

Thus, the simplified expression is:

\(\frac{9}{4}\)

Example 2: Simplifying with Variables

Consider the following expression:

\(\left( \frac{x}{y} \right)^3\)

Step 1: Distribute the exponent 3 to both the numerator and denominator:

\(\frac{x^3}{y^3}\)

Thus, the simplified expression is:

\(\frac{x^3}{y^3}\)

Practice Questions

  1. Question 1: Simplify the following expression:

    \(\left( \frac{4}{5} \right)^3\)

    Solution

    Step 1: Distribute the exponent 3 to both the numerator and denominator:

    \(\frac{4^3}{5^3}\)

    Step 2: Simplify:

    \(\frac{64}{125}\)

  2. Question 2: Simplify the following expression:

    \(\left( \frac{a}{b} \right)^4\)

    Solution

    Step 1: Distribute the exponent 4 to both the numerator and denominator:

    \(\frac{a^4}{b^4}\)

  3. Question 3: Simplify the following expression:

    \(\left( \frac{2x}{3y} \right)^2\)

    Solution

    Step 1: Distribute the exponent 2 to both the numerator and denominator:

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

    Step 2: Simplify:

    \(\frac{4x^2}{9y^2}\)

  4. Question 4: Simplify the following expression:

    \(\left( \frac{m}{n} \right)^5\)

    Solution

    Step 1: Distribute the exponent 5 to both the numerator and denominator:

    \(\frac{m^5}{n^5}\)

  5. Question 5: Simplify the following expression:

    \(\left( \frac{6}{7} \right)^2\)

    Solution

    Step 1: Distribute the exponent 2 to both the numerator and denominator:

    \(\frac{6^2}{7^2}\)

    Step 2: Simplify:

    \(\frac{36}{49}\)

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