If y = 3^k, which of the following expressions is equivalent to 9^{k+1} for all real value… — GMAT

If $y = 3^k$ , which of the following expressions is equivalent to $9^{k+1}$ for all real values of $k$ ?

The expression

9y^2

is equivalent to

9^{k+1}

Applying the product rule of exponents gives

9^{k+1} = 9^k \cdot 9^1

. Expressing 9 as

3^2

gives

9^k = (3^2)^k = (3^k)^2

. Substituting

y = 3^k

yields

9 \cdot y^2 = 9y^2

Rewrite

9^{k+1}

using the product rule of exponents (

a^{m+n} = a^m \cdot a^n

9^{k+1} = 9^k \cdot 9^1 = 9 \cdot 9^k

Separating the exponent addition into a product allows simplification of the constant factor.

Express the base 9 in terms of base 3.

9^k = (3^2)^k = (3^k)^2

Since

y = 3^k

, converting base 9 to

3^2

allows substitution of

y

into the expression.

Substitute

y = 3^k

into the expression.

9 \cdot (3^k)^2 = 9y^2

Replacing

3^k

with

y

yields the simplified algebraic expression in terms of

y

Properties of exponents including power of a power

(a^m)^n = a^{mn}

and product of powers

a^{m+n} = a^m \cdot a^n

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