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)
=(4^(-5)·4^5)·(9^5·9^(-3))
=4^0·9^2
=1·9^2
=9^2
=81

Explanation:

Step1: Apply power - of - a - product property

$(ab)^n=a^n\cdot b^n$, so $(4\cdot9)^5 = 4^5\cdot9^5$

Step2: Apply associative property of multiplication

$a\cdot(b\cdot c)=(a\cdot b)\cdot c$, so $4^{- 5}\cdot(4^5\cdot9^5)\cdot9^{-3}=(4^{-5}\cdot4^5)\cdot(9^5\cdot9^{-3})$

Step3: Apply product - of - powers property

$a^m\cdot a^n=a^{m + n}$, so $4^{-5}\cdot4^5=4^{-5 + 5}=4^0$ and $9^5\cdot9^{-3}=9^{5+( - 3)}=9^2$

Step4: Apply zero - exponent property

$a^0 = 1$ ($a
eq0$), so $4^0 = 1$

Step5: Apply multiplicative identity property

$1\cdot a=a$, so $1\cdot9^2=9^2$

Step6: Calculate $9^2$

$9^2=9\times9 = 81$

Answer:

The properties applied in order are: power - of - a - product property, associative property of multiplication, product - of - powers property, zero - exponent property, multiplicative identity property, and basic exponent calculation.