Which of the following is an equivalent expression to 15^0.7^-2 with only positive exponents that has been generated by applying the Properties of Integer Exponents?

Answer 1

To simplify the expression \( 15^{0.7^{-2}} \) using the Properties of Integer Exponents, we will first analyze \( 0.7^{-2} \).

1. Rewriting the Exponent: The expression \( a^{-b} = \frac{1}{a^b} \). Hence, we can rewrite \( 0.7^{-2} \) as: \[ 0.7^{-2} = \frac{1}{0.7^2} \]

2. Substituting: Now substituting this back into the expression gives: \[ 15^{0.7^{-2}} = 15^{\frac{1}{0.7^2}} \]

3. Breaking Down the Exponent: We can also express \( \frac{1}{a^b} \) as \( a^{-b} \). Thus we rewrite: \[ 15^{0.7^{-2}} = 15^{\frac{1}{(0.7^2)}} = 15^{(0.7^{-2})} \]

4. Evaluating \( 0.7^2 \): Next, let's calculate \( 0.7^2 \): \[ 0.7^2 = 0.49 \] Therefore, we now have: \[ 15^{\frac{1}{0.49}} = 15^{\frac{100}{49}} \quad (\text{if we multiply numerator and denominator by } 100) \]

5. Final Expression: We can write this in a different way or simply leave it as it is since \( 15^{\frac{100}{49}} \) respects positive exponents.

Thus, an equivalent expression to \( 15^{0.7^{-2}} \) that is generated by applying properties of integer exponents and has only positive exponents is: \[ 15^{\frac{100}{49}} \]