Questions: Divide and simplify to lowest terms. (d^3-a^3)/(d^2-a^2) ÷ (7d-7a)/(7d+7a) = □ (Simplify your answer.)

Solution Steps

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

1. Factorize the numerator and denominator of the first fraction.
2. Factorize the numerator and denominator of the second fraction.
3. Perform the division by multiplying the first fraction by the reciprocal of the second fraction.
4. Simplify the resulting expression by canceling out common factors.

We start with the expression:

$$\frac{d^{3}-a^{3}}{d^{2}-a^{2}} \div \frac{7 d-7 a}{7 d+7 a}$$

First, we factor the numerator and denominator of the first fraction:

• The numerator $d^{3} - a^{3}$ can be factored as:

$$d^{3} - a^{3} = -(a - d)(a^{2} + ad + d^{2})$$

• The denominator $d^{2} - a^{2}$ can be factored as:

$$d^{2} - a^{2} = -(a - d)(a + d)$$

Next, we factor the second fraction:

• The numerator $7d - 7a$ can be factored as:

$$7(d - a) = -7(a - d)$$

• The denominator $7d + 7a$ can be factored as:

$$7(d + a)$$

Now we rewrite the division of the two fractions as multiplication by the reciprocal:

$$\frac{-(a - d)(a^{2} + ad + d^{2})}{-(a - d)(a + d)} \times \frac{7(d + a)}{-7(a - d)}$$

We can simplify the expression by canceling out common factors:

$$= \frac{(a^{2} + ad + d^{2})}{(a + d)} \times \frac{(d + a)}{(a - d)}$$

This simplifies to:

$$= \frac{-(a^{2} + ad + d^{2})}{(a - d)}$$

Final Answer