Apply the Properties of Integer Exponents to generate equivalent expressions to (7−3)^7 ⋅ 7^20/6^−2

Question

with only positive exponents, then solve the expression. Provide the answer as an improper fraction.(1 point)
(7−3)^7 ⋅ 7^20/6^−2=

Answer 1

To simplify \((7^{-3})^7 \cdot 7^{20} / 6^{-2}\) using the properties of integer exponents, we will apply the rules of exponents step by step.

1. Apply the Power Rule: \((7^{-3})^7\) can be simplified using the power of a power property \((a^m)^n = a^{m \cdot n}\): \[ (7^{-3})^7 = 7^{-21} \]

2. Combine Exponents: Now we combine \(7^{-21}\) and \(7^{20}\) using the product of powers property \(a^m \cdot a^n = a^{m+n}\): \[ 7^{-21} \cdot 7^{20} = 7^{-21 + 20} = 7^{-1} \]

3. Rewrite the Expression: The expression now looks like: \[ \frac{7^{-1}}{6^{-2}} \]

4. Apply the Negative Exponent Rule: Recall the negative exponent rule \(a^{-m} = \frac{1}{a^m}\). Hence, \[ 7^{-1} = \frac{1}{7} \] and \[ 6^{-2} = \frac{1}{6^2} = \frac{1}{36} \]

5. Rewrite the Division: Therefore, we can rewrite our expression as: \[ \frac{\frac{1}{7}}{\frac{1}{36}} \]

6. Dividing Fractions: To divide by a fraction, we multiply by its reciprocal: \[ \frac{1}{7} \cdot 36 = \frac{36}{7} \]

Thus, the final answer is: \[ \boxed{\frac{36}{7}} \]