Question

1. vocabulary identify the property that is being applied at each step to simplify the expression. 4^(-5)·(4·9)^5·9^(-3)=4^(-5)·(4^5·9^5)·9^(-3) product property =(4^(-5)·4^5)·(9^5·9^(-3))

Explanation:

Step1: Identify the first - step property

The step $4^{-5}\cdot(4\cdot9)^{5}\cdot9^{-3}=4^{-5}\cdot(4^{5}\cdot9^{5})\cdot9^{-3}$ uses the power - of - a - product property $(ab)^n=a^n\cdot b^n$, where $a = 4$, $b = 9$ and $n = 5$.

Step2: Identify the second - step property

The step $4^{-5}\cdot(4^{5}\cdot9^{5})\cdot9^{-3}=(4^{-5}\cdot4^{5})\cdot(9^{5}\cdot9^{-3})$ uses the associative property of multiplication. The associative property of multiplication states that for real numbers $a$, $b$, and $c$, $(a\cdot b)\cdot c=a\cdot(b\cdot c)$. Here, we grouped the terms with the same base together.

Answer:

The first step uses the power - of - a - product property and the second step uses the associative property of multiplication.