Basic Math Examples

$\frac{(z - 2) (z + 7)}{z^{2} + 4} \div (z^{2} - 49)$

Step 1

Rewrite the division as a fraction.

$\frac{\frac{(z - 2) (z + 7)}{z^{2} + 4}}{z^{2} - 49}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{(z - 2) (z + 7)}{z^{2} + 4} \cdot \frac{1}{z^{2} - 49}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $49$ as $7^{2}$ .

$\frac{(z - 2) (z + 7)}{z^{2} + 4} \cdot \frac{1}{z^{2} - 7^{2}}$

Step 3.2

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 $b = 7$ .

$\frac{(z - 2) (z + 7)}{z^{2} + 4} \cdot \frac{1}{(z + 7) (z - 7)}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $z + 7$ .

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Step 4.1.1

Factor $z + 7$ out of $(z - 2) (z + 7)$ .

$\frac{(z + 7) (z - 2)}{z^{2} + 4} \cdot \frac{1}{(z + 7) (z - 7)}$

Step 4.1.2

Cancel the common factor.

$\frac{(z + 7) (z - 2)}{z^{2} + 4} \cdot \frac{1}{(z + 7) (z - 7)}$

Step 4.1.3

Rewrite the expression.

$\frac{z - 2}{z^{2} + 4} \cdot \frac{1}{z - 7}$

Step 4.2

Multiply $\frac{z - 2}{z^{2} + 4}$ by $\frac{1}{z - 7}$ .

$\frac{z - 2}{(z^{2} + 4) (z - 7)}$