Question

select the equivalent expression.
\\(\left(\dfrac{2^{-10}}{4^2}\
ight)^7=?\\)
choose 1 answer:
a \\(2^{-3}\cdot 4^{-9}\\)
b \\(2^{84}\\)
c \\(\dfrac{1}{2^{70}\cdot 4^{14}}\\)

Explanation:

Step1: Rewrite 4 as power of 2

$4^2 = (2^2)^2 = 2^{4}$

Step2: Simplify the base fraction

$\frac{2^{-10}}{2^{4}} = 2^{-10-4} = 2^{-14}$

Step3: Apply exponent to the base

$(2^{-14})^7 = 2^{-14 \times 7} = 2^{-98}$

Step4: Verify each option

Option A:

$2^{-3} \cdot 4^{-9} = 2^{-3} \cdot (2^2)^{-9} = 2^{-3} \cdot 2^{-18} = 2^{-21}
eq 2^{-98}$

Option B:

$2^{84}
eq 2^{-98}$

Option C:

$\frac{1}{2^{70} \cdot 4^{14}} = \frac{1}{2^{70} \cdot (2^2)^{14}} = \frac{1}{2^{70} \cdot 2^{28}} = \frac{1}{2^{98}} = 2^{-98}$

Answer:

C. $\frac{1}{2^{70} \cdot 4^{14}}$