Evaluate each expression. See Example 4. (-2)⁴

Verified step by step guidance

1. Identify the base and the exponent in the expression. Here, the base is \(-2\) and the exponent is \(4\).
2. Recall that raising a number to an exponent means multiplying the base by itself as many times as the exponent indicates. So, \((-2)^4 = (-2) \times (-2) \times (-2) \times (-2)\).
3. Multiply the base step-by-step: first multiply the first two factors, then multiply the result by the next factor, and so on.
4. Remember that multiplying two negative numbers results in a positive number, so keep track of the signs carefully during multiplication.
5. Continue multiplying until all four factors are included, which will give you the final value of \((-2)^4\).

Key Concepts

Exponents and Powers

An exponent indicates how many times a base number is multiplied by itself. For example, in (-2)⁴, the base is -2 and the exponent 4 means multiplying -2 by itself four times: (-2) × (-2) × (-2) × (-2).

Order of Operations with Negative Bases

When raising a negative number to a power, parentheses are crucial. (-2)⁴ means the entire negative number is raised to the fourth power, while -2⁴ means the negative of 2 raised to the fourth power. This affects the final sign of the result.

Even and Odd Powers of Negative Numbers

Raising a negative number to an even power results in a positive number because multiplying an even number of negatives yields a positive. Conversely, an odd power of a negative number results in a negative value.