Question
3. \( \frac{x^{2}-a^{2}}{(x+a)^{2}} \cdot \frac{2 x+2 a}{3 x}= \)
Ask by Gray Ball. in the United States
Feb 26,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The simplified expression is \( \frac{2(x - a)}{3x} \).
Solution
We start with the expression:
[(x² - a²) / (x + a)²] · [(2x + 2a) / (3x)]
Step 1: Factor the numerators where possible. Note that:
x² - a² = (x - a)(x + a)
2x + 2a = 2(x + a)
So the expression becomes:
[(x - a)(x + a) / (x + a)²] · [2(x + a) / (3x)]
Step 2: Simplify by canceling a common factor (x + a) in the first fraction:
(x + a) cancels with one (x + a) in the denominator, yielding:
[(x - a) / (x + a)] · [2(x + a) / (3x)]
Step 3: Now cancel the remaining (x + a) factor in the numerator of the second fraction with the (x + a) in the first fraction:
(x + a) cancels:
Result = (x - a) · 2 / (3x)
So the simplified expression is:
2(x - a) / (3x)
Remember that this cancellation is valid provided x + a ≠ 0 and x ≠ 0.
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Simplify this solution Beyond the Answer
To simplify the expression \( \frac{x^{2}-a^{2}}{(x+a)^{2}} \cdot \frac{2 x+2 a}{3 x} \), let's break it down step by step. First, recognize that \( x^2 - a^2 \) can be factored using the difference of squares: \[ x^2 - a^2 = (x - a)(x + a) \] Now, rewriting the original expression, we have: \[ \frac{(x - a)(x + a)}{(x + a)^2} \cdot \frac{2(x + a)}{3x} \] Next, we can simplify by cancelling the \( (x + a) \) terms in the fractions. This gives us: \[ \frac{(x - a)}{(x + a)} \cdot \frac{2}{3} \] Now, let's combine these: \[ \frac{2(x - a)}{3(x + a)} \] Thus, the simplified expression is: \[ \frac{2(x - a)}{3(x + a)} \]
