Pre-Algebra Examples

$\frac{z^{2} + 11 z + 18}{z^{2} + 12 z + 27} \div \frac{z^{2} + 2 z}{z^{2} \cdot (5 z) + 6}$

Step 1

Factor $z^{2} + 11 z + 18$ using the AC method.

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Step 1.1

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 $18$ and whose sum is $11$ .

$2, 9$

Step 1.2

Write the factored form using these integers.

$\frac{(z + 2) (z + 9)}{z^{2} + 12 z + 27} \div \frac{z^{2} + 2 z}{z^{2} \cdot (5 z) + 6}$

Step 2

Factor $z^{2} + 12 z + 27$ using the AC method.

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Step 2.1

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 $27$ and whose sum is $12$ .

$3, 9$

Step 2.2

Write the factored form using these integers.

$\frac{(z + 2) (z + 9)}{(z + 3) (z + 9)} \div \frac{z^{2} + 2 z}{z^{2} \cdot (5 z) + 6}$

Step 3

Reduce the expression $\frac{(z + 2) (z + 9)}{(z + 3) (z + 9)}$ by cancelling the common factors.

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Step 3.1

Cancel the common factor.

$\frac{(z + 2) (z + 9)}{(z + 3) (z + 9)} \div \frac{z^{2} + 2 z}{z^{2} \cdot (5 z) + 6}$

Step 3.2

Rewrite the expression.

$\frac{z + 2}{z + 3} \div \frac{z^{2} + 2 z}{z^{2} \cdot (5 z) + 6}$

Step 4

Factor $z$ out of $z^{2} + 2 z$ .

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Step 4.1

Factor $z$ out of $z^{2}$ .

$\frac{z + 2}{z + 3} \div \frac{z \cdot z + 2 z}{z^{2} \cdot (5 z) + 6}$

Step 4.2

Factor $z$ out of $2 z$ .

$\frac{z + 2}{z + 3} \div \frac{z \cdot z + z \cdot 2}{z^{2} \cdot (5 z) + 6}$

Step 4.3

Factor $z$ out of $z \cdot z + z \cdot 2$ .

$\frac{z + 2}{z + 3} \div \frac{z (z + 2)}{z^{2} \cdot (5 z) + 6}$

Step 5

Rewrite using the commutative property of multiplication.

$\frac{z + 2}{z + 3} \div \frac{z (z + 2)}{5 z^{2} z + 6}$

Step 6

Multiply $z^{2}$ by $z$ by adding the exponents.

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Step 6.1

Move $z$ .

$\frac{z + 2}{z + 3} \div \frac{z (z + 2)}{5 (z \cdot z^{2}) + 6}$

Step 6.2

Multiply $z$ by $z^{2}$ .

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Step 6.2.1

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

$\frac{z + 2}{z + 3} \div \frac{z (z + 2)}{5 (z^{1} z^{2}) + 6}$

Step 6.2.2

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

$\frac{z + 2}{z + 3} \div \frac{z (z + 2)}{5 z^{1 + 2} + 6}$

Step 6.3

Add $1$ and $2$ .

$\frac{z + 2}{z + 3} \div \frac{z (z + 2)}{5 z^{3} + 6}$