Simplify (y^2-w^2)/(y^3-w^3)*(y^2+yw+w^2)/(y^2+2yw+w^2)

$\frac{y^{2} - w^{2}}{y^{3} - w^{3}} \cdot \frac{y^{2} + y w + w^{2}}{y^{2} + 2 y w + w^{2}}$

Step 1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = w$ .

$\frac{(y + w) (y - w)}{y^{3} - w^{3}} \cdot \frac{y^{2} + y w + w^{2}}{y^{2} + 2 y w + w^{2}}$

Step 2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = y$ and $b = w$ .

$\frac{(y + w) (y - w)}{(y - w) (y^{2} + y w + w^{2})} \cdot \frac{y^{2} + y w + w^{2}}{y^{2} + 2 y w + w^{2}}$

Step 3

Factor using the perfect square rule.

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 y w = 2 \cdot y \cdot w$

Rewrite the polynomial.

$\frac{(y + w) (y - w)}{(y - w) (y^{2} + y w + w^{2})} \cdot \frac{y^{2} + y w + w^{2}}{y^{2} + 2 \cdot y \cdot w + w^{2}}$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = y$ and $b = w$ .

$\frac{(y + w) (y - w)}{(y - w) (y^{2} + y w + w^{2})} \cdot \frac{y^{2} + y w + w^{2}}{{(y + w)}^{2}}$

Step 4

Simplify terms.

Cancel the common factor of $y + w$ .

Factor $y + w$ out of ${(y + w)}^{2}$ .

$\frac{(y + w) (y - w)}{(y - w) (y^{2} + y w + w^{2})} \cdot \frac{y^{2} + y w + w^{2}}{(y + w) (y + w)}$

Cancel the common factor.

$\frac{(y + w) (y - w)}{(y - w) (y^{2} + y w + w^{2})} \cdot \frac{y^{2} + y w + w^{2}}{(y + w) (y + w)}$

Rewrite the expression.

$\frac{y - w}{(y - w) (y^{2} + y w + w^{2})} \cdot \frac{y^{2} + y w + w^{2}}{y + w}$

Cancel the common factor of $y^{2} + y w + w^{2}$ .

Factor $y^{2} + y w + w^{2}$ out of $(y - w) (y^{2} + y w + w^{2})$ .

$\frac{y - w}{(y^{2} + y w + w^{2}) (y - w)} \cdot \frac{y^{2} + y w + w^{2}}{y + w}$

Cancel the common factor.

$\frac{y - w}{(y^{2} + y w + w^{2}) (y - w)} \cdot \frac{y^{2} + y w + w^{2}}{y + w}$

Rewrite the expression.

$\frac{y - w}{y - w} \cdot \frac{1}{y + w}$

Multiply $\frac{y - w}{y - w}$ by $\frac{1}{y + w}$ .

$\frac{y - w}{(y - w) (y + w)}$

Cancel the common factor of $y - w$ .

Cancel the common factor.

$\frac{y - w}{(y - w) (y + w)}$

Rewrite the expression.

$\frac{1}{y + w}$

Simplify (y^2-w^2)/(y^3-w^3)*(y^2+yw+w^2)/(y^2+2yw+w^2)

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