x\textsuperscript{2}y Step 2: \frac{x\textsuperscript{2}y}{y-2x\textsuperscript{2}} 1 Step 3: \frac{y-2x\textsuperscript{2}}{x\textsuperscript{2}y} \cdot \frac{1}{y-2x\textsuperscript{2}} What should Mrs. Cho do next? Find a common denominator for the two fractions. Divide the numerator and denominator of the fraction. Multiply the numerators, multiply the denominators. Multiply the first fraction by the reciprocal of the second fraction. Mark this and return
x\textsuperscript{2}y Step 2: \frac{x\textsuperscript{2}y}{y-2x\textsuperscript{2}} 1 Step 3: \frac{y-2x\textsuperscript{2}}{x\textsuperscript{2}y} \cdot \frac{1}{y-2x\textsuperscript{2}} What should Mrs. Cho do next? Find a common denominator for the two fractions. Divide the numerator and denominator of the fraction. Multiply the numerators, multiply the denominators. Multiply the first fraction by the reciprocal of the second fraction. Mark this and return
The Correct Answer and Explanation is:
Correct Answer: Multiply the numerators, multiply the denominators.
Explanation
In the given problem, Mrs. Cho is simplifying a complex algebraic expression involving rational expressions. Let’s break down the steps:
- Step 1: x2yx^2y (likely the original expression or part of it)
- Step 2: x2yy−2×2\frac{x^2y}{y – 2x^2}
- Step 3: y−2x2x2y⋅1y−2×2\frac{y – 2x^2}{x^2y} \cdot \frac{1}{y – 2x^2}
At Step 3, she is multiplying two fractions:
- y−2x2x2y\frac{y – 2x^2}{x^2y}
- 1y−2×2\frac{1}{y – 2x^2}
When multiplying two fractions, the correct procedure is to multiply the numerators together and multiply the denominators together. This is a fundamental rule of fraction multiplication, and it applies to algebraic fractions just as it does to numerical fractions.
So, applying this rule: y−2x2x2y⋅1y−2×2=(y−2×2)⋅1x2y⋅(y−2×2)=y−2x2x2y(y−2×2)\frac{y – 2x^2}{x^2y} \cdot \frac{1}{y – 2x^2} = \frac{(y – 2x^2) \cdot 1}{x^2y \cdot (y – 2x^2)} = \frac{y – 2x^2}{x^2y(y – 2x^2)}
Now, we can simplify by canceling out the common term y−2x2y – 2x^2 in the numerator and denominator: =1x2y= \frac{1}{x^2y}
This is the simplified result.
Therefore, the appropriate next step for Mrs. Cho is to multiply the numerators and multiply the denominators, followed by simplification. The other options are not suitable here:
- “Find a common denominator” is for addition/subtraction, not multiplication.
- “Divide the numerator and denominator of the fraction” is vague and not relevant here.
- “Multiply the first fraction by the reciprocal of the second” is used for division, not multiplication.
