Questions: Perform the operation indicated. Simplify. (x+2y)/(24x^2) * (3x)/(x^2+xy-2y^2) 1/(8x(x-y)) 8x(x-y) 8x/(x-y) 1/(8x(x+y))

Transcript text: Perform the operation indicated. Simplify. \[ \frac{x+2 y}{24 x^{2}} \cdot \frac{3 x}{x^{2}+x y-2 y^{2}} \] $\frac{1}{8 x(x-y)}$ $8 x(x-y)$ $\frac{8 x}{(x-y)}$ $\frac{1}{8 x(x+y)}$

Solution

Solution Steps

To simplify the given expression, we need to follow these steps:

Factorize the denominators and numerators where possible.
Cancel out common factors in the numerator and the denominator.
Simplify the resulting expression.

Step 1: Expression Setup

We start with the expression: \[ \frac{x + 2y}{24x^2} \cdot \frac{3x}{x^2 + xy - 2y^2} \]

Step 2: Simplification

First, we can rewrite the expression as: \[ \frac{(x + 2y) \cdot 3x}{24x^2 \cdot (x^2 + xy - 2y^2)} \] Next, we simplify the numerator and denominator. The numerator becomes: \[ 3x(x + 2y) \] And the denominator is: \[ 24x^2(x^2 + xy - 2y^2) \]

Step 3: Factorization

We can factor the denominator $x^2 + xy - 2y^2$. The expression simplifies to: \[ \frac{3x(x + 2y)}{24x^2(x - y)(x + 2y)} \] Here, we notice that $x + 2y$ cancels out from the numerator and denominator.

Step 4: Final Simplification

After canceling $x + 2y$, we have: \[ \frac{3}{24x(x - y)} = \frac{1}{8x(x - y)} \]