Question

tyler simplified the expression \\(x^{-3} y^{9}\\). his procedure is shown below.

\\x^{-3} y^{9} = \frac{1}{x^{3}} \cdot \frac{1}{y^{-9}} = \frac{1}{x^{3} y^{-9}}\\

what is tylers error?

• both powers should be in the numerator with positive exponents.
• both powers should be in the denominator with positive exponents.
• the power \\(x^{3}\\) should be in the numerator and the power \\(y^{9}\\) in the denominator.
• the power \\(y^{9}\\) should be in the numerator and the power \\(x^{3}\\) in the denominator.

Explanation:

Analyze the given expression and Tyler's work

$$ x^{-3} y^9 = \frac{1}{x^3} \cdot \frac{1}{y^{-9}} = \frac{1}{x^3 y^{-9}} $$

Identify the correct simplification using exponent rules

$$ x^{-3} y^9 = \frac{1}{x^3} \cdot y^9 = \frac{y^9}{x^3} $$

Determine Tyler's error

$$ \text{Tyler incorrectly wrote } y^9 \text{ as } \frac{1}{y^{-9}} \text{ in the denominator, keeping a negative exponent.} \text{Correctly, the power } y^9 \text{ should be in the numerator and the power } x^3 \text{ in the denominator.} $$

Answer:

• Both powers should be in the numerator with positive exponents.
• Both powers should be in the denominator with positive exponents.
• The power \(x^3\) should be in the numerator and the power \(y^9\) in the denominator.
• The power \(y^9\) should be in the numerator and the power \(x^3\) in the denominator. (Correct answer)