Question

Which expressions are equivalent to \dfrac{4^{-3}}{4^{-1}} 4 −1 4 −3 ​ start fraction, 4, start superscript, minus, 3, end superscript, divided by, 4, start superscript, minus, 1, end superscript, end fraction ? Choose 2 answers: Choose 2 answers: (Choice A) A \dfrac{4^1}{4^3} 4 3 4 1 ​ start fraction, 4, start superscript, 1, end superscript, divided by, 4, cubed, end fraction (Choice B) B \dfrac{1}{4^{2}} 4 2 1 ​ start fraction, 1, divided by, 4, squared, end fraction (Choice C) C 4^3\cdot 4^14 3 ⋅4 1 4, cubed, dot, 4, start superscript, 1, end superscript (Choice D) D (4^{-1})^{-3}(4 −1 ) −3

Answer 1

Answer: 4^-2

Explanation:

Answer 2

`(4^(-3))/(4^(-1)) = (4^(1))/(4^(3))`

`(4^(-3))/(4^(-1)) = (1)/(4^(2))`

Explanation:

Given

`(4^(-3))/(4^(-1))`

Required

Choose equivalent expressions

Choosing the first answer:

`(4^(-3))/(4^(-1))`

Split expressions

`4^(-3) * (1)/(4^(-1))`

Apply laws of indices:
`(a^(-b) = (1)/(a^b))`

`(1)/(4^3) * (1)/(4^(-1))`

Apply laws of indices:
`(a^(-b) = (1)/(a^b))`

`(1)/(4^3) * (1)/(1/4)`

`(1)/(4^3) * (4^1)/(1)`

`(4^1)/(4^3)`

Hence:

`(4^(-3))/(4^(-1)) = (4^(1))/(4^(3))`

Choosing the second:

`(4^(-3))/(4^(-1))`

Apply law of indices:
`((a^m)/(a^n) = a^(m-n))`

So,

`(4^(-3))/(4^(-1)) = 4^(-3-(-1))`

`(4^(-3))/(4^(-1)) = 4^(-3+1))`

`(4^(-3))/(4^(-1)) = 4^(-2)`

Apply law of indices:
`(a^(-b) = (1)/(a^b))`

So:

`(4^(-3))/(4^(-1)) = (1)/(4^(2))`