Problem 4 - Algebra 1: Concepts and Skills

Question

Use the quotient of powers property to simplify the expression. $$ \frac{7^{6}}{7^{9}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{1}{343} \).

Step 1: Identify the quotient of powers property

The quotient of powers property is \( \frac{a^n}{a^m} = a^{n-m} \). This is useful when we have the same base (a) being raised to a power in the numerator and another power in the denominator. Here, our base (a) = 7, n = 6 and m = 9.

Step 2: Apply the quotient of powers property

Use the quotient of powers property to simplify \( \frac{7^{6}}{7^{9}} \). Using the property \( \frac{a^n}{a^m} = a^{n-m} \), this simplifies to \( 7^{6-9} \).

Step 3: Simplify further

Then perform the subtraction in the exponent, \( 7^{6-9} = 7^{-3} \). This represents 1 divided by 7 raised to the 3rd power.

Step 4: Final simplification

Simplify \( 7^{-3} \) as \( \frac{1}{7^3} \), which equals \( \frac{1}{343} \).