Evaluate each exponential expression in Exercises 1–22. 4⁻³

Verified step by step guidance

1. Identify the base and the exponent in the expression \$4^{-3}$. Here, the base is 4 and the exponent is -3.
2. Recall the rule for negative exponents: \(a^{-n} = \frac{1}{a^n}\), where \(a\) is a nonzero number and \(n\) is a positive integer.
3. Apply the negative exponent rule to rewrite \$4^{-3}$ as \(\frac{1}{4^3}\).
4. Calculate the positive exponent part \$4^3$ by multiplying 4 by itself three times: \(4 \times 4 \times 4\).
5. Express the final answer as \(\frac{1}{4^3}\), which is \(\frac{1}{64}\) after evaluating the multiplication.

Key Concepts

Exponents and Powers

Exponents indicate how many times a base number is multiplied by itself. For example, 4^3 means 4 × 4 × 4. Understanding this helps in evaluating expressions with positive exponents.

Negative Exponents

A negative exponent means taking the reciprocal of the base raised to the corresponding positive exponent. For instance, 4^(-3) equals 1 divided by 4^3, which is 1/64.

Evaluating Exponential Expressions

To evaluate an exponential expression, calculate the power indicated by the exponent, considering the sign of the exponent. This involves applying rules for both positive and negative exponents to simplify the expression.