3. $$ \frac{x^{2}-a^{2}}{(x+a)^{2}} \cdot \frac{2 x+2 a}{3 x}= $$

Question

UpStudy AI · Accepted Answer

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.
The simplified expression is $$ \frac{2(x - a)}{3x} $$.

3. \( \frac{x^{2}-a^{2}}{(x+a)^{2}} \cdot \frac{2 x+2 a}{3 x}= \)