Problem 117

To determine which pairs of numbers are opposite to each other, we analyze each option step by step:

Option A:

- Calculate the square root of $(-2)^2$: $\sqrt{{(-2)}^2} = \sqrt{4} = 2$.
- Compare $-2$ and $2$: These are opposite numbers because the definition of opposite numbers is two numbers that have the same magnitude but different signs.
- Therefore, $-2$ and $\sqrt{{(-2)}^2}$ are opposite to each other.

Option B:

- Calculate the cube root of $-8$: $\sqrt[3]{-8} = -2$.
- Compare $-2$ and $-2$: These numbers are equal, not opposite to each other because opposite numbers must have different signs.
- Therefore, $-2$ and $\sqrt[3]{-8}$ are not opposite to each other.

Option C:

- Compare $-2$ and $-\frac{1}{2}$: These are not opposite numbers. They are reciprocals, not opposites, because reciprocals are numbers whose product is $1$, and opposite numbers are numbers with the same magnitude but opposite signs.
- Therefore, $-2$ and $-\frac{1}{2}$ are not opposite to each other.

Option D:

- Calculate the absolute value of $-2$: $|-2| = 2$.
- Compare $2$ and $2$: These numbers are equal, not opposite to each other because opposite numbers must have different signs.
- Therefore, $2$ and $|-2|$ are not opposite to each other.

Based on the analysis, the correct answer is:

$\boxed{A}$