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

Verified step by step guidance

1. Recognize that the expression \((-2)^4\) means raising the number \(-2\) to the power of 4, which involves multiplying \(-2\) by itself 4 times.
2. Write the expression as \((-2) \times (-2) \times (-2) \times (-2)\) to visualize the repeated multiplication.
3. Recall the rule that multiplying two negative numbers results in a positive number, so pair the factors to simplify: \(((-2) \times (-2)) \times ((-2) \times (-2))\).
4. Calculate each pair: \((-2) \times (-2) = 4\), so the expression becomes \(4 \times 4\).
5. Multiply the results of the pairs to get the final value: \(4 \times 4\).

Key Concepts

Order of Operations and Exponentiation

Understanding the order of operations is crucial when evaluating expressions with exponents. Exponentiation is performed before multiplication or addition, so (-2)⁴ means raising -2 to the fourth power, not just 2 to the fourth power.

Properties of Exponents with Negative Bases

When a negative number is raised to an even power, the result is positive because multiplying an even number of negative factors results in a positive product. For example, (-2)⁴ = (-2) × (-2) × (-2) × (-2) = 16.

Evaluating Powers of Integers

Evaluating powers involves repeated multiplication of the base by itself. For integer exponents, this means multiplying the base as many times as the exponent indicates, which helps in simplifying expressions like (-2)⁴ accurately.