Basic Math Examples

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

Step 1

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 $a = z$ and $b = y$ .

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

Step 2

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

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

Step 3

Reduce the expression $\frac{(z - y) (z^{2} + z y + y^{2})}{(z + y) (z - y)}$ by cancelling the common factors.

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{{(z + y)}^{2}}{z - y} \cdot (\frac{z^{2} + z y + y^{2}}{z + y} \cdot \frac{1}{{(z - y)}^{2}})$

Step 5

Multiply $\frac{z^{2} + z y + y^{2}}{z + y}$ by $\frac{1}{{(z - y)}^{2}}$ .

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

Step 6

Cancel the common factor of $z + y$ .

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

Cancel the common factor.

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

Step 6.4

Rewrite the expression.

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

Step 7

Multiply $\frac{z + y}{z - y}$ by $\frac{z^{2} + z y + y^{2}}{{(z - y)}^{2}}$ .

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

Step 8

Multiply $z - y$ by ${(z - y)}^{2}$ by adding the exponents.

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Step 8.1

Multiply $z - y$ by ${(z - y)}^{2}$ .

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Step 8.1.1

Raise $z - y$ to the power of $1$ .

$\frac{(z + y) (z^{2} + z y + y^{2})}{{(z - y)}^{1} {(z - y)}^{2}}$

Step 8.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(z + y) (z^{2} + z y + y^{2})}{{(z - y)}^{1 + 2}}$

Step 8.2

Add $1$ and $2$ .

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