(1.) $$ a) \frac{a^{2}-a b}{a^{2}+a b} \cdot \frac{a^{2} b+a b^{2}}{a b}=\frac{(a-b)}{a(a+b)}=a b $$

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UpStudy AI · Accepted Answer

Let's solve the equation step by step.

We start with the left-hand side of the equation:

$$
LHS = \frac{a^{2}-ab}{a^{2}+ab} \cdot \frac{a^{2}b+ab^{2}}{ab}
$$

### Step 1: Simplify the first fraction

The first fraction is:

$$
\frac{a^{2}-ab}{a^{2}+ab}
$$

We can factor out $$a$$ from both the numerator and the denominator:

$$
= \frac{a(a-b)}{a(a+b)} = \frac{a-b}{a+b}
$$

### Step 2: Simplify the second fraction

The second fraction is:

$$
\frac{a^{2}b + ab^{2}}{ab}
$$

We can factor out $$ab$$ from the numerator:

$$
= \frac{ab(a+b)}{ab} = a+b
$$

### Step 3: Combine the simplified fractions

Now we can combine the simplified fractions:

$$
LHS = \frac{a-b}{a+b} \cdot (a+b)
$$

The $$a+b$$ in the numerator and denominator cancels out:

$$
LHS = a-b
$$

### Step 4: Set the left-hand side equal to the right-hand side

Now we set the left-hand side equal to the right-hand side:

$$
a-b = \frac{(a-b)}{a(a+b)}
$$

### Step 5: Analyze the equation

The equation simplifies to:

$$
a-b = ab
$$

### Conclusion

Thus, we have shown that:

$$
\frac{a^{2}-ab}{a^{2}+ab} \cdot \frac{a^{2}b+ab^{2}}{ab} = a-b
$$

This confirms that the left-hand side equals the right-hand side under the condition that $$a-b = ab$$.

If you have any further questions or need additional assistance, feel free to ask!
$$ \frac{a^{2}-ab}{a^{2}+ab} \cdot \frac{a^{2}b+ab^{2}}{ab} = a - b $$

(1.) \( a) \frac{a^{2}-a b}{a^{2}+a b} \cdot \frac{a^{2} b+a b^{2}}{a b}=\frac{(a-b)}{a(a+b)}=a b \)

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