Algebra Examples

$\frac{2 v}{3 v^{3} - 9 v^{2}} \div \frac{v - 8}{v^{2} - 11 v + 24}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{2 v}{3 v^{3} - 9 v^{2}} \cdot \frac{v^{2} - 11 v + 24}{v - 8}$

Step 2

Simplify terms.

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Step 2.1

Factor $3 v^{2}$ out of $3 v^{3} - 9 v^{2}$ .

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Step 2.1.1

Factor $3 v^{2}$ out of $3 v^{3}$ .

$\frac{2 v}{3 v^{2} (v) - 9 v^{2}} \cdot \frac{v^{2} - 11 v + 24}{v - 8}$

Step 2.1.2

Factor $3 v^{2}$ out of $- 9 v^{2}$ .

$\frac{2 v}{3 v^{2} (v) + 3 v^{2} (- 3)} \cdot \frac{v^{2} - 11 v + 24}{v - 8}$

Step 2.1.3

Factor $3 v^{2}$ out of $3 v^{2} (v) + 3 v^{2} (- 3)$ .

$\frac{2 v}{3 v^{2} (v - 3)} \cdot \frac{v^{2} - 11 v + 24}{v - 8}$

Step 2.2

Cancel the common factor of $v$ and $v^{2}$ .

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Step 2.2.1

Factor $v$ out of $2 v$ .

$\frac{v \cdot 2}{3 v^{2} (v - 3)} \cdot \frac{v^{2} - 11 v + 24}{v - 8}$

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Factor $v$ out of $3 v^{2} (v - 3)$ .

$\frac{v \cdot 2}{v (3 v (v - 3))} \cdot \frac{v^{2} - 11 v + 24}{v - 8}$

Step 2.2.2.2

Cancel the common factor.

$\frac{v \cdot 2}{v (3 v (v - 3))} \cdot \frac{v^{2} - 11 v + 24}{v - 8}$

Step 2.2.2.3

Rewrite the expression.

$\frac{2}{3 v (v - 3)} \cdot \frac{v^{2} - 11 v + 24}{v - 8}$

Step 3

Factor $v^{2} - 11 v + 24$ 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 $24$ and whose sum is $- 11$ .

$- 8, - 3$

Step 3.2

Write the factored form using these integers.

$\frac{2}{3 v (v - 3)} \cdot \frac{(v - 8) (v - 3)}{v - 8}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $v - 3$ .

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Step 4.1.1

Factor $v - 3$ out of $3 v (v - 3)$ .

$\frac{2}{(v - 3) (3 v)} \cdot \frac{(v - 8) (v - 3)}{v - 8}$

Step 4.1.2

Factor $v - 3$ out of $(v - 8) (v - 3)$ .

$\frac{2}{(v - 3) (3 v)} \cdot \frac{(v - 3) (v - 8)}{v - 8}$

Step 4.1.3

Cancel the common factor.

$\frac{2}{(v - 3) (3 v)} \cdot \frac{(v - 3) (v - 8)}{v - 8}$

Step 4.1.4

Rewrite the expression.

$\frac{2}{3 v} \cdot \frac{v - 8}{v - 8}$

Step 4.2

Multiply $\frac{2}{3 v}$ by $\frac{v - 8}{v - 8}$ .

$\frac{2 (v - 8)}{3 v (v - 8)}$

Step 4.3

Cancel the common factor of $v - 8$ .

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Step 4.3.1

Cancel the common factor.

$\frac{2 (v - 8)}{3 v (v - 8)}$

Step 4.3.2

Rewrite the expression.

$\frac{2}{3 v}$