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 (42^(-7))/(42^(-2))=42^(-7+2)=42^(-5)\checkmark`

#C

`\\ \rm\Rrightarrow (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((1)/(42^(-5))\right)^(-1)=42^(-5)\checkmark`

#f

`\\ \rm\Rrightarrow \left((6^(-5))/(7^5)\right)^(-1)`

`\\ \rm\Rrightarrow (42^(-5))^(-1)`

`\\ \rm\Rrightarrow 42^5`

Answer 2

B, C, E

Explanation:

Exponent Rules

`a^b \cdot a^c=a^(b+c)`

`(a^b)/(a^c)=a^(b-c)`

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

`a^0=1`

`(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 (42^(-7))/(42^(-2))=42^(-7-(-2))=42^(-5)`

`\textsf{C}\quad (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((1)/(42^(-5))\right)^(-1)=(42^5)^(-1)=42^(-5)`

`\textsf{F}\quad \left((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`