GMAT Exponents: If a > 0, b = 2a, and c = 2b,…

If a > 0, b = 2a, and c = 2b, which of the following is equivalent to a^{b} b^{c} ?

A. $(2 a^{4})^{a}$
B. $(2 a^{6})^{a}$
C. $(2 a^{8})^{a}$
D. $(16 a^{6})^{a}$
E. $(16 a^{8})^{a}$

exponentsexpression formulas

Explanation	Calculations	Help
$We first express the given relationships of b and c in terms of a$	$b = 2 a c = 2 b$
$We substitute these expressions for b and c into a^{b} b^{c}$	$a^{b} \times b^{c} = a^{2 a} \times (2 a)^{4 a}$
$We expand (2 a)^{4 a} by raising each factor to the power 4 a$	$(2 a)^{4 a} = 2^{4 a} \times a^{4 a}$
$We apply Multiplying Exponents of the Same Base to combine a^{2 a} and a^{4 a}$	$a^{2 a} \times a^{4 a} = a^{2 a + 4 a} = a^{6 a}$
$We rewrite 2^{4 a} and a^{6 a} using Exponents of Exponents and combine them$	$2^{4 a} = (2^{4})^{a} = 1 6^{a} a^{6 a} = (a^{6})^{a} 2^{4 a} \times a^{6 a} = (16 \times a^{6})^{a}$

(16a^6)^a (D)

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