Full Prep Academy

Overview

The Quotient of Powers rule states that when dividing powers with the same base, you subtract the exponents. In mathematical terms:

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

This rule applies only when the base is the same in both terms. The exponents are subtracted, with the numerator's exponent being the larger one.

Steps to Apply the Quotient of Powers Rule

1. Identify the common base in both terms.
2. Subtract the exponent of the denominator from the exponent of the numerator.
3. Write the base and the result of the subtraction as a single power.

Example 1: Simplifying with Same Base

Consider the following expression:

\( \frac{5^8}{5^3} \)

Step 1: The base is \( 5 \), so we subtract the exponents 8 and 3:

\( 5^{(8-3)} = 5^5 \)

Thus, the simplified expression is:

\( 5^5 \)

Example 2: Simplifying with Variables

Consider the following expression:

\( \frac{x^7}{x^4} \)

Step 1: The base is \( x \), so we subtract the exponents 7 and 4:

\( x^{(7-4)} = x^3 \)

Thus, the simplified expression is:

\( x^3 \)

Practice Questions

1. Question 1: Simplify the following expression:

   \( \frac{3^9}{3^4} \)

   Solution

   Step 1: The base is \( 3 \), so we subtract the exponents 9 and 4:

   \( 3^{(9-4)} = 3^5 \)

2. Question 2: Simplify the following expression:

   \( \frac{y^{10}}{y^2} \)

   Solution

   Step 1: The base is \( y \), so we subtract the exponents 10 and 2:

   \( y^{(10-2)} = y^8 \)

3. Question 3: Simplify the following expression:

   \( \frac{7^5}{7^3} \)

   Solution

   Step 1: The base is \( 7 \), so we subtract the exponents 5 and 3:

   \( 7^{(5-3)} = 7^2 \)

4. Question 4: Simplify the following expression:

   \( \frac{a^{12}}{a^6} \)

   Solution

   Step 1: The base is \( a \), so we subtract the exponents 12 and 6:

   \( a^{(12-6)} = a^6 \)

5. Question 5: Simplify the following expression:

   \( \frac{8^7}{8^4} \)

   Solution

   Step 1: The base is \( 8 \), so we subtract the exponents 7 and 4:

   \( 8^{(7-4)} = 8^3 \)