Question

Which expressions are equivalent to 42-5? Choose all the correct answers. Show your work in arriving at your answers.

Answer 1

Check one by one

#A

$\\ \rm\Rrightarrow 42^{-3}42^8=42^{-3+8}=42^5$

#B

$\\ \rm\Rrightarrow \dfrac{42^{-7}}{42^{-2}}=42^{-7+2}=42^{-5}\checkmark$

#C

$\\ \rm\Rrightarrow \dfrac{6^{-5}}{7^5}=6^{-5}7^{-5}=42^{-5}\checkmark$

#d

$\\ \rm\Rrightarrow 42^0.42^5=42^{0+5}=42^5$

#e

$\\ \rm\Rrightarrow \left(\dfrac{1}{42^{-5}}\right)^{-1}=42^{-5}\checkmark$

#f

$\\ \rm\Rrightarrow \left(\dfrac{6^{-5}}{7^5}\right)^{-1}$

$\\ \rm\Rrightarrow (42^{-5})^{-1}$

$\\ \rm\Rrightarrow 42^5$

Answer 2

Answer:

B, C, E

Step-by-step explanation:

Exponent Rules

$a^b \cdot a^c=a^{b+c}$

$\dfrac{a^b}{a^c}=a^{b-c}$

$a^n \cdot b^n=(ab)^n$

$a^0=1$

$\dfrac{1}{a^b}=a^{-b}$

$(a^b)^c=a^{bc}$

$\textsf{A}\quad42^{-3} \cdot 42^8=42^{-3+8}=42^5$

$\textsf{B}\quad \dfrac{42^{-7}}{42^{-2}}=42^{-7-(-2)}=42^{-5}$

$\textsf{C}\quad \dfrac{6^{-5}}{7^5}=6^{-5} \cdot 7^{-5}=(6 \cdot 7)^{-5}=42^{-5}$

$\textsf{D}\quad 42^0 \cdot 42^5=1 \cdot 42^5=42^5$

$\textsf{E}\quad \left(\dfrac{1}{42^{-5}}\right)^{-1}=(42^5)^{-1}=42^{-5}$

$\textsf{F}\quad \left(\dfrac{6^{-5}}{7^5}\right)^{-1}=\left(6^{-5} \cdot 7^{-5}\right)^{-1}=\left(\left(6 \cdot 7\right)^{-5}\right)^{-1}=42^{-5(-1)}=42^5$