Evaluate each exponential expression in Exercises 1–22. −2⁶

Verified step by step guidance

1. Identify the expression given: \(-2^6\). Notice that the exponent applies only to the base 2, not to the negative sign, because there are no parentheses around the -2.
2. Recall the order of operations: exponents are evaluated before multiplication or applying the negative sign. So first, calculate \$2^6$.
3. Calculate \$2^6$ by multiplying 2 by itself 6 times: \(2 \times 2 \times 2 \times 2 \times 2 \times 2\).
4. After finding the value of \$2^6$, apply the negative sign in front of the result, which means multiply the result by -1.
5. Write the final expression as \(- (2^6)\) and simplify to get the value of the original expression.

Key Concepts

Order of Operations

The order of operations dictates the sequence in which mathematical operations are performed. Exponents are evaluated before multiplication or negation. In the expression −2^6, the exponent applies only to 2, and the negative sign is applied afterward.

Exponents

Exponents represent repeated multiplication of a base number. For example, 2^6 means multiplying 2 by itself six times (2 × 2 × 2 × 2 × 2 × 2). Understanding how to compute powers is essential for evaluating exponential expressions.

Negative Sign and Exponents

A negative sign in front of a base with an exponent can change the result depending on parentheses. Without parentheses, −2^6 means the negative of 2^6, resulting in −64. With parentheses, (−2)^6 means raising −2 to the sixth power, resulting in a positive value.