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 the expression \((7^{-3})^7 \cdot \frac{7^{20}}{6^{-2}}\) using the properties of integer exponents, we can follow these steps:

1. Apply the power of a power rule: \[ (a^m)^n = a^{m \cdot n} \] Therefore, \[ (7^{-3})^7 = 7^{-3 \cdot 7} = 7^{-21} \]

2. Combine the powers of \(7\): \[ 7^{-21} \cdot 7^{20} = 7^{-21 + 20} = 7^{-1} \]

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

4. Convert negative exponents to positive: Remember that \(a^{-m} = \frac{1}{a^m}\). Thus: \[ 7^{-1} = \frac{1}{7} \quad \text{and} \quad 6^{-2} = \frac{1}{6^2} = \frac{1}{36} \]

5. Substitute and simplify: Therefore, the expression becomes: \[ \frac{\frac{1}{7}}{\frac{1}{36}} = \frac{1}{7} \cdot 36 = \frac{36}{7} \]

So, the final answer in the form of an improper fraction is: \[ \frac{36}{7} \]

Thus, \[ (7^{-3})^7 \cdot \frac{7^{20}}{6^{-2}} = \frac{36}{7} \]