2015 AMC 10B Problem 1

P1-10~1 minPrint

Exponents, Logarithms & Radicals Set up the equation/formula and compute; no special trick needed

What is the value of $2-(-2)^{-2}$ ?

Show solution without answering

Key insight

A negative power means one over the positive power, and squaring -2 gives positive 4. So the expression is 2 minus 1/4.

Solution

Do the power first, then the subtraction.

A negative exponent means "one over". So $(-2)^{-2}$ is one over $(-2)^2$ .
Now square: $(-2)^2 = (-2) \cdot (-2) = 4$ . A negative times a negative is positive, so the minus sign is gone.
So $(-2)^{-2} = \dfrac{1}{4}$ .
Finally, $2 - \dfrac14 = \dfrac84 - \dfrac14 = \dfrac74$ .

The answer is $\boxed{\textbf{(C)}\ \frac{7}{4}}$ .

Why this works

Two rules do all the work. First, $a^{-n}$ is the same as $\dfrac{1}{a^n}$ . Second, an even power of a negative number is positive. Because of the parentheses, the square applies to the whole $-2$ , sign included, so check for parentheses before deciding what sign a power has.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

Handle the power first. A negative exponent is a reciprocal:
$(-2)^{-2} = \frac{1}{(-2)^2} = \frac{1}{4},$
and the square of $-2$ is positive $4$ , so the sign disappears.

Then
$2 - \frac{1}{4} = \frac{8}{4} - \frac{1}{4} = \frac{7}{4}.$

The answer is $\boxed{\textbf{(C)}\ \frac{7}{4}}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 1: Follow the Rules (§1.6, §1.7, §1.8)
Introduction to Algebra, Chapter 11: Special Factorizations (§11.4)
Introduction to Algebra, Chapter 19: Exponents and Logarithms (§19.4)