Question

_16. which expressions are equivalent to \\((xy^{-8})^3\\) for all values of x and y where the expression is defined?

select two correct answers.

a. \\(xy^{-24}\\)
b. \\(x^3y^{-5}\\)
c. \\(\frac{x^3}{y^5}\\)
d. \\(\frac{x^3}{y^{24}}\\)
e. \\(\frac{x}{y^{24}}\\)
f. \\(x^3y^{-24}\\)

Explanation:

Simplify the given expression

Using the Exponent Rules and Algebraic Simplification knowledge points

$$ (xy^{-8})^3 = x^3 \cdot (y^{-8})^3 = x^3 y^{-24} $$

Rewrite with positive exponents

Using the Exponent Rules and Algebraic Simplification knowledge points

$$ x^3 y^{-24} = \frac{x^3}{y^{24}} $$

Evaluate the given options

Using the Algebraic Simplification knowledge point

• Option A: \(xy^{-24}\) (Incorrect)
• Option B: \(x^3y^{-5}\) (Incorrect)
• Option C: \(\frac{x^3}{y^5}\) (Incorrect)
• Option D: \(\frac{x^3}{y^{24}}\) (Correct)
• Option E: \(\frac{x}{y^{24}}\) (Incorrect)
• Option F: \(x^3y^{-24}\) (Correct)

Answer:

• A. \(xy^{-24}\)
• B. \(x^3y^{-5}\)
• C. \(\frac{x^3}{y^5}\)
• D. \(\frac{x^3}{y^{24}}\) (Correct answer)
• E. \(\frac{x}{y^{24}}\)
• F. \(x^3y^{-24}\) (Correct answer)