If 2^{2x+2} - 2^{2x} = 12 for some real number x, what is the value of 2^{4x}? — SAT

If $2^{2x+2} - 2^{2x} = 12$ for some real number $x$ , what is the value of $2^{4x}$ ?

Answer: 16

By applying the product rule of exponents, the expression

2^{2x+2}

can be rewritten as

2^{2x} \cdot 2^2

, or

4 \cdot 2^{2x}

. Substituting this into the given equation yields

4 \cdot 2^{2x} - 2^{2x} = 12

. Factoring out

2^{2x}

gives

2^{2x}(4 - 1) = 12

, which simplifies to

3 \cdot 2^{2x} = 12

. Dividing both sides of the equation by 3 results in

2^{2x} = 4

. Since

2^{4x}

can be written as

(2^{2x})^2

, substituting 4 for

2^{2x}

gives

4^2 = 16

. Alternatively, solving

2^{2x} = 4

gives

2x = 2

, which means

x = 1

. Substituting

x = 1

into

2^{4x}

yields

2^{4(1)} = 2^4 = 16

Rewrite

2^{2x+2}

using exponent properties.

2^{2x+2} = 2^{2x} \cdot 2^2 = 4 \cdot 2^{2x}

To express both exponential terms with the same base power, allowing them to be factored or combined.

Substitute this back into the equation and factor out the common term

2^{2x}

2^{2x}(4 - 1) = 12 \implies 3 \cdot 2^{2x} = 12

To isolate the exponential expression.

Solve for

2^{2x}

by dividing both sides of the equation by 3.

2^{2x} = 4

To find the value of the exponential term.

Express

2^{4x}

in terms of

2^{2x}

and evaluate.

2^{4x} = (2^{2x})^2 = 4^2 = 16

To find the final requested value using the exponent rule

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

Factoring exponential equations and applying power of a power exponent rules

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