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: $$\boxed{{\displaystyle \frac{36}{7}}}$$