Basic Math Examples

$\frac{\frac{\frac{z + 5}{z - 6}}{z^{2} + 4 z - 12}}{z^{2} - 8 z + 12}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{\frac{z + 5}{z - 6}}{z^{2} + 4 z - 12} \cdot \frac{1}{z^{2} - 8 z + 12}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{z + 5}{z - 6} \cdot \frac{1}{z^{2} + 4 z - 12} \frac{1}{z^{2} - 8 z + 12}$

Step 3

Factor $z^{2} + 4 z - 12$ using the AC method.

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Step 3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 12$ and whose sum is $4$ .

$- 2, 6$

Step 3.2

Write the factored form using these integers.

$\frac{z + 5}{z - 6} \cdot \frac{1}{(z - 2) (z + 6)} \frac{1}{z^{2} - 8 z + 12}$

Step 4

Multiply $\frac{z + 5}{z - 6}$ by $\frac{1}{(z - 2) (z + 6)}$ .

$\frac{z + 5}{(z - 6) ((z - 2) (z + 6))} \cdot \frac{1}{z^{2} - 8 z + 12}$

Step 5

Factor $z^{2} - 8 z + 12$ using the AC method.

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Step 5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $12$ and whose sum is $- 8$ .

$- 6, - 2$

Step 5.2

Write the factored form using these integers.

$\frac{z + 5}{(z - 6) (z - 2) (z + 6)} \cdot \frac{1}{(z - 6) (z - 2)}$

Step 6

Multiply $\frac{z + 5}{(z - 6) (z - 2) (z + 6)} \cdot \frac{1}{(z - 6) (z - 2)}$ .

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Step 6.1

Multiply $\frac{z + 5}{(z - 6) (z - 2) (z + 6)}$ by $\frac{1}{(z - 6) (z - 2)}$ .

$\frac{z + 5}{(z - 6) (z - 2) (z + 6) ((z - 6) (z - 2))}$

Step 6.2

Raise $z - 6$ to the power of $1$ .

$\frac{z + 5}{{(z - 6)}^{1} (z - 6) (z - 2) (z + 6) (z - 2)}$

Step 6.3

Raise $z - 6$ to the power of $1$ .

$\frac{z + 5}{{(z - 6)}^{1} {(z - 6)}^{1} (z - 2) (z + 6) (z - 2)}$

Step 6.4

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

$\frac{z + 5}{{(z - 6)}^{1 + 1} (z - 2) (z + 6) (z - 2)}$

Step 6.5

Add $1$ and $1$ .

$\frac{z + 5}{{(z - 6)}^{2} (z - 2) (z + 6) (z - 2)}$

Step 6.6

Raise $z - 2$ to the power of $1$ .

$\frac{z + 5}{{(z - 6)}^{2} ({(z - 2)}^{1} (z - 2)) (z + 6)}$

Step 6.7

Raise $z - 2$ to the power of $1$ .

$\frac{z + 5}{{(z - 6)}^{2} ({(z - 2)}^{1} {(z - 2)}^{1}) (z + 6)}$

Step 6.8

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

$\frac{z + 5}{{(z - 6)}^{2} {(z - 2)}^{1 + 1} (z + 6)}$

Step 6.9

Add $1$ and $1$ .

$\frac{z + 5}{{(z - 6)}^{2} {(z - 2)}^{2} (z + 6)}$