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

Question

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Respuesta :

MrRoyal MrRoyal · Answer 1 · 2020-10-12T02:50:27+02:00

Answer:

[tex]\dfrac{4^{-3}}{4^{-1}} = \dfrac{4^{1}}{4^{3}}[/tex]

[tex]\dfrac{4^{-3}}{4^{-1}} = \dfrac{1}{4^{2}}[/tex]

Step-by-step explanation:

Given

[tex]\dfrac{4^{-3}}{4^{-1}}[/tex]

Required

Choose equivalent expressions

Choosing the first answer:

[tex]\dfrac{4^{-3}}{4^{-1}}[/tex]

Split expressions

[tex]4^{-3} * \frac{1}{4^{-1}}[/tex]

Apply laws of indices: [tex](a^{-b} = \frac{1}{a^b})[/tex]

[tex]\frac{1}{4^3} * \frac{1}{4^{-1}}[/tex]

Apply laws of indices: [tex](a^{-b} = \frac{1}{a^b})[/tex]

[tex]\frac{1}{4^3} * \frac{1}{1/4}[/tex]

[tex]\frac{1}{4^3} * \frac{4^1}{1}[/tex]

[tex]\frac{4^1}{4^3}[/tex]

Hence:

[tex]\dfrac{4^{-3}}{4^{-1}} = \dfrac{4^{1}}{4^{3}}[/tex]

Choosing the second:

[tex]\dfrac{4^{-3}}{4^{-1}}[/tex]

Apply law of indices: [tex](\frac{a^m}{a^n} = a^{m-n})[/tex]

So,

[tex]\dfrac{4^{-3}}{4^{-1}} = 4^{-3-(-1)}[/tex]

[tex]\dfrac{4^{-3}}{4^{-1}} = 4^{-3+1)}[/tex]

[tex]\dfrac{4^{-3}}{4^{-1}} = 4^{-2}[/tex]

Apply law of indices: [tex](a^{-b} = \frac{1}{a^b})[/tex]

So:

[tex]\dfrac{4^{-3}}{4^{-1}} = \dfrac{1}{4^{2}}[/tex]