Evaluate each exponential expression in Exercises 1–22. (−2)^6

Verified step by step guidance

1. Identify the base and the exponent in the expression \((-2)^6\).
2. Understand that the base is \(-2\) and the exponent is \(6\), which means you need to multiply \(-2\) by itself 6 times.
3. Write the expression as \((-2) \times (-2) \times (-2) \times (-2) \times (-2) \times (-2)\).
4. Recall that multiplying two negative numbers results in a positive number, and multiplying a positive number by a negative number results in a negative number.
5. Calculate the product step by step, keeping track of the sign changes: first pair \((-2) \times (-2) = 4\), then \(4 \times (-2) = -8\), and continue this process until all factors are multiplied.

Key Concepts

Exponential Expressions

Exponential expressions involve a base raised to a power, indicating 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 how to evaluate these expressions is crucial, as it involves both multiplication and the properties of exponents.

Negative Bases

When dealing with negative bases, such as (-2), the evaluation of the expression can yield different results depending on whether the exponent is even or odd. An even exponent results in a positive value, while an odd exponent results in a negative value. This distinction is important for accurately calculating the value of expressions with negative bases.

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. In evaluating expressions, one must follow the order: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right). This principle is essential for correctly solving exponential expressions.