Verified Answer :

  • Rewrite 64 as a power of 4, specifically \(64 = 4^3\).
  • Apply the rule \((a^m)^n = a^{m \times n}\) to simplify \((4^3)^{\frac{2}{3}}\) to \(4^2\).
  • Calculate the final value \(4^2 = 16\).
  • Select the correct option which is \(\boxed{16}\).

Explanation

  1. Analyzing the Expression

The expression to evaluate is \(64^{\frac{2}{3}}\). This involves a base raised to a fractional exponent. The properties of exponents allow us to simplify this expression by recognizing that the base \(64\) can be written as a power of a simpler integer. We note that \(64 = 4^3\). Therefore, the expression can be rewritten as \((4^3)^{\frac{2}{3}}\).

  1. Applying Exponent Rules

To simplify \((4^3)^{\frac{2}{3}}\), we use the power of a power rule for exponents, which states that \((a^m)^n = a^{m \times n}\). Applying this rule, we multiply the exponents: \(3 \times \frac{2}{3}\). This results in \(3 \times \frac{2}{3} = 2\). Thus, the expression simplifies to \(4^2\).

  1. Final Calculation and Conclusion

Finally, we compute \(4^2\), which is \(4 \times 4 = 16\). Comparing this result with the provided options (64, 4, 16, 32), we identify that 16 is the correct answer.

Examples

Understanding how to work with fractional exponents is a fundamental skill in algebra, useful in fields like finance and physics. For example, in compound interest formulas, you might encounter fractional powers to determine values over periods that aren't whole years. Mastering these exponent rules allows for quick and accurate simplification of complex mathematical models used in daily tasks and further scientific studies.

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