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

Verified step by step guidance

1. Identify the base and the exponent in the expression \((-2)^4\). Here, the base is \(-2\) and the exponent is \(4\).
2. Understand that the exponent \(4\) indicates that the base \(-2\) should be multiplied by itself a total of four times.
3. Write the expression as a repeated multiplication: \((-2) \times (-2) \times (-2) \times (-2)\).
4. Calculate the product of the first two terms: \((-2) \times (-2) = 4\).
5. Continue multiplying the result by the next \(-2\) until all terms are used: \(4 \times (-2) = -8\) and then \(-8 \times (-2)\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The exponent indicates how many times the base is multiplied by itself. For example, in the expression a^n, 'a' is the base and 'n' is the exponent. Understanding this concept is crucial for evaluating expressions like (-2)⁴, where the base is negative.

Negative Base

When dealing with a negative base raised to an exponent, the sign of the result depends on whether the exponent is even or odd. If the exponent is even, the result is positive, while if it is odd, the result is negative. In the case of (-2)⁴, since 4 is even, the outcome will be positive, illustrating the importance of recognizing the base's sign.

Order of Operations

The order of operations is a set of rules that dictates the sequence in which mathematical operations should be performed to ensure consistent results. The common acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) helps remember this order. In evaluating (-2)⁴, the exponentiation is performed before any other operations, which is essential for arriving at the correct answer.