If k is a positive integer, which of the following expressions is equivalent to \frac{2^{k… — GRE General Test

k

is a positive integer, which of the following expressions is equivalent to

\frac{2^{k+2} \cdot 3^{2k-1} + 6^k \cdot 3^{k-1}}{12^k \cdot 2^{-1}}

for all values of

k

The expression simplifies to

\frac{10}{3} \left(\frac{3}{2}\right)^k

Factoring the terms in the numerator into base

18^k

gives

\frac{4}{3} \cdot 18^k + \frac{1}{3} \cdot 18^k = \frac{5}{3} \cdot 18^k

. The denominator equals

\frac{1}{2} \cdot 12^k

. Dividing numerator by denominator yields

\frac{5/3}{1/2} \cdot \left(\frac{18}{12}\right)^k = \frac{10}{3} \left(\frac{3}{2}\right)^k

Rewrite each term in the numerator using prime base factorization

2^{k+2} \cdot 3^{2k-1} = 2^k \cdot 2^2 \cdot (3^2)^k \cdot 3^{-1} = 4 \cdot 2^k \cdot 9^k \cdot \frac{1}{3} = \frac{4}{3} \cdot 18^k

, and

6^k \cdot 3^{k-1} = (2 \cdot 3)^k \cdot 3^k \cdot 3^{-1} = 2^k \cdot 9^k \cdot \frac{1}{3} = \frac{1}{3} \cdot 18^k

Converting all powers to base 18 allows terms with identical exponential factors to be combined.

Combine the terms in the numerator

\frac{4}{3} \cdot 18^k + \frac{1}{3} \cdot 18^k = \left(\frac{4}{3} + \frac{1}{3}\right) \cdot 18^k = \frac{5}{3} \cdot 18^k

Adding coefficients of like exponential terms.

Simplify the denominator expression

12^k \cdot 2^{-1} = \frac{1}{2} \cdot 12^k

Applying the negative exponent rule

a^{-n} = \frac{1}{a^n}

Divide the numerator by the denominator

\frac{\frac{5}{3} \cdot 18^k}{\frac{1}{2} \cdot 12^k} = \frac{5/3}{1/2} \cdot \left(\frac{18}{12}\right)^k = \left(\frac{5}{3} \cdot 2\right) \cdot \left(\frac{3}{2}\right)^k = \frac{10}{3} \left(\frac{3}{2}\right)^k

Dividing fractions by multiplying by the reciprocal and applying quotient rule for powers with the same exponent.

Prime base factorization and laws of exponents

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