2014 AMC 10B Problem 2

The fraction has a top and a bottom. Each is small enough to work out as ordinary numbers, so do that first.

1. Top. $2^3 = 8$, so $2^3 + 2^3 = 8 + 8 = 16$.
2. Bottom. A negative exponent means "one over": $2^{-3} = \frac{1}{2^3} = \frac18$. So $2^{-3} + 2^{-3} = \frac18 + \frac18 = \frac28 = \frac14$.
3. Divide. Dividing by $\frac14$ is the same as multiplying by $4$:
$$\frac{16}{1/4} = 16 \times 4 = 64.$$

The answer is $\boxed{\textbf{(E)}\ 64}$.

Adding a number to itself doubles it, and doubling a power of $2$ just raises the exponent by one: $8 + 8 = 16$ is $2^3 + 2^3 = 2^4$. The mistake to avoid is adding the exponents when you add the numbers; exponents add only when you multiply. When the numbers are small, turning each power into an ordinary number is the safest route.

Stay with powers of $2$ the whole time. The top is $2 \cdot 2^3 = 2^4$ and the bottom is $2 \cdot 2^{-3} = 2^{-2}$. Dividing powers of the same base subtracts the exponents: $2^{4 - (-2)} = 2^{6} = 64$.

The numerator is $2^3 + 2^3 = 8 + 8 = 16$. The denominator is $2^{-3} + 2^{-3} = \frac18 + \frac18 = \frac14$.

Dividing, $\frac{16}{1/4} = 16 \cdot 4 = 64$.

Equivalently, $2^3 + 2^3 = 2 \cdot 2^3 = 2^4$ and $2^{-3} + 2^{-3} = 2^{-2}$, so the quotient is $2^{4 - (-2)} = 2^6 = 64$.

The answer is $\boxed{\textbf{(E)}\ 64}$.