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:

$\dfrac{4^{-3}}{4^{-1}} = \dfrac{4^{1}}{4^{3}}$

$\dfrac{4^{-3}}{4^{-1}} = \dfrac{1}{4^{2}}$

Step-by-step explanation:

Given

$\dfrac{4^{-3}}{4^{-1}}$

Required

Choose equivalent expressions

Choosing the first answer:

$\dfrac{4^{-3}}{4^{-1}}$

Split expressions

$4^{-3} * \frac{1}{4^{-1}}$

Apply laws of indices: $(a^{-b} = \frac{1}{a^b})$

$\frac{1}{4^3} * \frac{1}{4^{-1}}$

Apply laws of indices: $(a^{-b} = \frac{1}{a^b})$

$\frac{1}{4^3} * \frac{1}{1/4}$

$\frac{1}{4^3} * \frac{4^1}{1}$

$\frac{4^1}{4^3}$

Hence:

$\dfrac{4^{-3}}{4^{-1}} = \dfrac{4^{1}}{4^{3}}$

Choosing the second:

$\dfrac{4^{-3}}{4^{-1}}$

Apply law of indices: $(\frac{a^m}{a^n} = a^{m-n})$

So,

$\dfrac{4^{-3}}{4^{-1}} = 4^{-3-(-1)}$

$\dfrac{4^{-3}}{4^{-1}} = 4^{-3+1)}$

$\dfrac{4^{-3}}{4^{-1}} = 4^{-2}$

Apply law of indices: $(a^{-b} = \frac{1}{a^b})$

So:

$\dfrac{4^{-3}}{4^{-1}} = \dfrac{1}{4^{2}}$