If x is a real number satisfying the exponential equation 9^{x+1} - 3^{2x+1} = 162, what i… — GRE General Test

If $x$ is a real number satisfying the exponential equation $9^{x+1} - 3^{2x+1} = 162$ , what is the value of $4^x$ ?

By converting

9^{x+1}

3^{2x+2}

and factoring out

3^{2x}

, the equation simplifies to

6 \cdot 3^{2x} = 162

. Dividing by 6 gives

3^{2x} = 27

, so

2x = 3

and

x = \frac{3}{2}

. Raising 4 to the power of

\frac{3}{2}

yields

(\sqrt{4})^3 = 8

Rewrite terms with a common base of 3.

9^{x+1} = (3^2)^{x+1} = 3^{2(x+1)} = 3^{2x+2}

Since 9 is a power of 3 (

9 = 3^2

), applying exponent rules converts the equation to base 3.

Factor out the common exponential expression

3^{2x}

3^{2x+2} - 3^{2x+1} = 3^{2x} \cdot 3^2 - 3^{2x} \cdot 3^1 = 3^{2x}(9 - 3) = 6 \cdot 3^{2x}

Using product rule of exponents (

3^{a+b} = 3^a \cdot 3^b

) allows combining like terms.

Solve for

x

6 \cdot 3^{2x} = 162 \implies 3^{2x} = 27 \implies 3^{2x} = 3^3 \implies 2x = 3 \implies x = \frac{3}{2}

Dividing both sides by 6 yields

3^{2x} = 27

, and equating exponents of matching bases gives

x = \frac{3}{2}

Evaluate the target expression

4^x

4^{3/2} = (4^{1/2})^3 = 2^3 = 8

Substituting

x = \frac{3}{2}

into

4^x

means taking the square root of 4 and raising it to the third power.

Solving exponential equations using common bases and exponent properties

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