Basic Math Examples

$\frac{z^{2} - 49}{z^{2} - 14 z + 49}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{z^{2} - 7^{2}}{z^{2} - 14 z + 49}$

Step 1.2

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 $b = 7$ .

$\frac{(z + 7) (z - 7)}{z^{2} - 14 z + 49}$

Step 2

Factor using the perfect square rule.

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Step 2.1

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

$\frac{(z + 7) (z - 7)}{z^{2} - 14 z + 7^{2}}$

Step 2.2

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

$14 z = 2 \cdot z \cdot 7$

Step 2.3

Rewrite the polynomial.

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

Step 2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = z$ and $b = 7$ .

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

Step 3

Cancel the common factor of $z - 7$ and ${(z - 7)}^{2}$ .

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Step 3.1

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

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

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $z - 7$ out of ${(z - 7)}^{2}$ .

$\frac{(z - 7) (z + 7)}{(z - 7) (z - 7)}$

Step 3.2.2

Cancel the common factor.

$\frac{(z - 7) (z + 7)}{(z - 7) (z - 7)}$

Step 3.2.3

Rewrite the expression.

$\frac{z + 7}{z - 7}$