2015 AMC 10A Problem 1

Follow the order of operations: first the powers inside the parentheses, then the sum, then the outside power, then the multiplication.

1. Work out the two powers inside. Any nonzero number to the power $0$ is $1$, so $2^0 = 1$. Also $5^2 = 25$.
2. Add up the parentheses:
$$1 - 1 + 25 + 0 = 25.$$
3. A power of $-1$ means "flip the number over" (take its reciprocal). So $25^{-1} = \frac{1}{25}$.
4. Multiply by $5$:
$$\frac{1}{25} \times 5 = \frac{5}{25} = \frac{1}{5}.$$

The answer is $\boxed{\textbf{(C)}\ \frac{1}{5}}$.

The problem checks two rules about exponents: a power of $0$ gives $1$, and a power of $-1$ gives the reciprocal, not a negative number. If you do the steps in the right order and keep those two rules straight, nothing else can go wrong. Notice that two wrong choices, $-125$ and $-120$, are exactly what you get by treating $-1$ as a minus sign.

Work inside the parentheses first. Since $2^0 = 1$ and $5^2 = 25$,
$$2^0 - 1 + 5^2 + 0 = 1 - 1 + 25 + 0 = 25.$$
The exponent $-1$ turns $25$ into its reciprocal $\frac{1}{25}$. Multiplying by $5$:
$$\frac{1}{25} \times 5 = \frac{5}{25} = \frac{1}{5}.$$

The answer is $\boxed{\textbf{(C)}\ \frac{1}{5}}$.