If 2^{a+b} = 12 and 2^{a-b} = 3, what is the value of 2^{2a} - 2^{2b}? — GMAT

If $2^{a+b} = 12$ and $2^{a-b} = 3$ , what is the value of $2^{2a} - 2^{2b}$ ?

Answer: 32

Multiplying the equations gives

(2^{a+b})(2^{a-b}) = 2^{2a} = 12 \times 3 = 36

. Dividing the first equation by the second gives

\frac{2^{a+b}}{2^{a-b}} = 2^{2b} = \frac{12}{3} = 4

. Subtracting the two values gives

2^{2a} - 2^{2b} = 36 - 4 = 32

Multiply the given exponent expressions to evaluate

2^{2a}

2^{2a} = (2^{a+b})(2^{a-b}) = 12 \times 3 = 36

According to the product rule of exponents,

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

. Adding

(a+b) + (a-b)

yields

2a

Divide the first exponent expression by the second to evaluate

2^{2b}

2^{2b} = \frac{2^{a+b}}{2^{a-b}} = \frac{12}{3} = 4

According to the quotient rule of exponents,

\frac{2^{x}}{2^{y}} = 2^{x-y}

. Subtracting

(a+b) - (a-b)

yields

2b

Calculate the value of

2^{2a} - 2^{2b}

36 - 4 = 32

Substitute the evaluated components

2^{2a} = 36

and

2^{2b} = 4

into the expression.

Exponent Operations (Product and Quotient Rules)

Estimated Time:1m 30s

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