Algebra Examples

$\frac{6 x^{3} - 600 x}{x^{2} - 18 x + 80} \cdot \frac{4 x}{x^{3} - 64 x} \div \frac{x + 10}{x^{3} - 64 x}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{6 x^{3} - 600 x}{x^{2} - 18 x + 80} \cdot \frac{4 x}{x^{3} - 64 x} \frac{x^{3} - 64 x}{x + 10}$

Step 2

Simplify the numerator.

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Step 2.1

Factor $6 x$ out of $6 x^{3} - 600 x$ .

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Step 2.1.1

Factor $6 x$ out of $6 x^{3}$ .

$\frac{6 x (x^{2}) - 600 x}{x^{2} - 18 x + 80} \cdot \frac{4 x}{x^{3} - 64 x} \frac{x^{3} - 64 x}{x + 10}$

Step 2.1.2

Factor $6 x$ out of $- 600 x$ .

$\frac{6 x (x^{2}) + 6 x (- 100)}{x^{2} - 18 x + 80} \cdot \frac{4 x}{x^{3} - 64 x} \frac{x^{3} - 64 x}{x + 10}$

Step 2.1.3

Factor $6 x$ out of $6 x (x^{2}) + 6 x (- 100)$ .

$\frac{6 x (x^{2} - 100)}{x^{2} - 18 x + 80} \cdot \frac{4 x}{x^{3} - 64 x} \frac{x^{3} - 64 x}{x + 10}$

Step 2.2

Rewrite $100$ as $10^{2}$ .

$\frac{6 x (x^{2} - 10^{2})}{x^{2} - 18 x + 80} \cdot \frac{4 x}{x^{3} - 64 x} \frac{x^{3} - 64 x}{x + 10}$

Step 2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 10$ .

$\frac{6 x (x + 10) (x - 10)}{x^{2} - 18 x + 80} \cdot \frac{4 x}{x^{3} - 64 x} \frac{x^{3} - 64 x}{x + 10}$

Step 3

Factor $x^{2} - 18 x + 80$ 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 $80$ and whose sum is $- 18$ .

$- 10, - 8$

Step 3.2

Write the factored form using these integers.

$\frac{6 x (x + 10) (x - 10)}{(x - 10) (x - 8)} \cdot \frac{4 x}{x^{3} - 64 x} \frac{x^{3} - 64 x}{x + 10}$

Step 4

Simplify the denominator.

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Step 4.1

Factor $x$ out of $x^{3} - 64 x$ .

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Step 4.1.1

Factor $x$ out of $x^{3}$ .

$\frac{6 x (x + 10) (x - 10)}{(x - 10) (x - 8)} \cdot \frac{4 x}{x \cdot x^{2} - 64 x} \frac{x^{3} - 64 x}{x + 10}$

Step 4.1.2

Factor $x$ out of $- 64 x$ .

$\frac{6 x (x + 10) (x - 10)}{(x - 10) (x - 8)} \cdot \frac{4 x}{x \cdot x^{2} + x \cdot - 64} \frac{x^{3} - 64 x}{x + 10}$

Step 4.1.3

Factor $x$ out of $x \cdot x^{2} + x \cdot - 64$ .

$\frac{6 x (x + 10) (x - 10)}{(x - 10) (x - 8)} \cdot \frac{4 x}{x (x^{2} - 64)} \frac{x^{3} - 64 x}{x + 10}$

Step 4.2

Rewrite $64$ as $8^{2}$ .

$\frac{6 x (x + 10) (x - 10)}{(x - 10) (x - 8)} \cdot \frac{4 x}{x (x^{2} - 8^{2})} \frac{x^{3} - 64 x}{x + 10}$

Step 4.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 8$ .

$\frac{6 x (x + 10) (x - 10)}{(x - 10) (x - 8)} \cdot \frac{4 x}{x (x + 8) (x - 8)} \frac{x^{3} - 64 x}{x + 10}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $x$ .

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Step 5.1.1

Factor $x$ out of $6 x (x + 10) (x - 10)$ .

$\frac{x ((6 (x + 10)) (x - 10))}{(x - 10) (x - 8)} \cdot \frac{4 x}{x (x + 8) (x - 8)} \frac{x^{3} - 64 x}{x + 10}$

Step 5.1.2

Factor $x$ out of $x (x + 8) (x - 8)$ .

$\frac{x ((6 (x + 10)) (x - 10))}{(x - 10) (x - 8)} \cdot \frac{4 x}{x ((x + 8) (x - 8))} \frac{x^{3} - 64 x}{x + 10}$

Step 5.1.3

Cancel the common factor.

$\frac{x ((6 (x + 10)) (x - 10))}{(x - 10) (x - 8)} \cdot \frac{4 x}{x ((x + 8) (x - 8))} \frac{x^{3} - 64 x}{x + 10}$

Step 5.1.4

Rewrite the expression.

$\frac{(6 (x + 10)) (x - 10)}{(x - 10) (x - 8)} \cdot \frac{4 x}{(x + 8) (x - 8)} \frac{x^{3} - 64 x}{x + 10}$

Step 5.2

Multiply $\frac{(6 (x + 10)) (x - 10)}{(x - 10) (x - 8)}$ by $\frac{4 x}{(x + 8) (x - 8)}$ .

$\frac{(6 (x + 10)) (x - 10) (4 x)}{(x - 10) (x - 8) ((x + 8) (x - 8))} \cdot \frac{x^{3} - 64 x}{x + 10}$

Step 5.3

Multiply $4$ by $6$ .

$\frac{24 (x + 10) (x - 10) x}{(x - 10) (x - 8) ((x + 8) (x - 8))} \cdot \frac{x^{3} - 64 x}{x + 10}$

Step 6

Raise $x - 8$ to the power of $1$ .

$\frac{24 (x + 10) (x - 10) x}{(x - 10) ({(x - 8)}^{1} (x - 8)) (x + 8)} \cdot \frac{x^{3} - 64 x}{x + 10}$

Step 7

Raise $x - 8$ to the power of $1$ .

$\frac{24 (x + 10) (x - 10) x}{(x - 10) ({(x - 8)}^{1} {(x - 8)}^{1}) (x + 8)} \cdot \frac{x^{3} - 64 x}{x + 10}$

Step 8

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

$\frac{24 (x + 10) (x - 10) x}{(x - 10) {(x - 8)}^{1 + 1} (x + 8)} \cdot \frac{x^{3} - 64 x}{x + 10}$

Step 9

Simplify terms.

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Step 9.1

Add $1$ and $1$ .

$\frac{24 (x + 10) (x - 10) x}{(x - 10) {(x - 8)}^{2} (x + 8)} \cdot \frac{x^{3} - 64 x}{x + 10}$

Step 9.2

Cancel the common factor of $x + 10$ .

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Step 9.2.1

Factor $x + 10$ out of $24 (x + 10) (x - 10) x$ .

$\frac{(x + 10) ((24 (x - 10)) x)}{(x - 10) {(x - 8)}^{2} (x + 8)} \cdot \frac{x^{3} - 64 x}{x + 10}$

Step 9.2.2

Cancel the common factor.

$\frac{(x + 10) ((24 (x - 10)) x)}{(x - 10) {(x - 8)}^{2} (x + 8)} \cdot \frac{x^{3} - 64 x}{x + 10}$

Step 9.2.3

Rewrite the expression.

$\frac{(24 (x - 10)) x}{(x - 10) {(x - 8)}^{2} (x + 8)} (x^{3} - 64 x)$

Step 9.3

Cancel the common factor of $x - 10$ .

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Step 9.3.1

Cancel the common factor.

$\frac{24 (x - 10) x}{(x - 10) {(x - 8)}^{2} (x + 8)} (x^{3} - 64 x)$

Step 9.3.2

Rewrite the expression.

$\frac{(24) x}{{(x - 8)}^{2} (x + 8)} (x^{3} - 64 x)$

Step 9.4

Multiply $\frac{24 x}{{(x - 8)}^{2} (x + 8)}$ by $x^{3} - 64 x$ .

$\frac{24 x (x^{3} - 64 x)}{{(x - 8)}^{2} (x + 8)}$

Step 10

Simplify the numerator.

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Step 10.1

Factor $x$ out of $x^{3} - 64 x$ .

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Step 10.1.1

Factor $x$ out of $x^{3}$ .

$\frac{24 x (x \cdot x^{2} - 64 x)}{{(x - 8)}^{2} (x + 8)}$

Step 10.1.2

Factor $x$ out of $- 64 x$ .

$\frac{24 x (x \cdot x^{2} + x \cdot - 64)}{{(x - 8)}^{2} (x + 8)}$

Step 10.1.3

Factor $x$ out of $x \cdot x^{2} + x \cdot - 64$ .

$\frac{24 x (x (x^{2} - 64))}{{(x - 8)}^{2} (x + 8)}$

Step 10.2

Rewrite $64$ as $8^{2}$ .

$\frac{24 x (x (x^{2} - 8^{2}))}{{(x - 8)}^{2} (x + 8)}$

Step 10.3

Factor.

$\frac{24 x \cdot x (x + 8) (x - 8)}{{(x - 8)}^{2} (x + 8)}$

Step 10.4

Combine exponents.

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Step 10.4.1

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

$\frac{24 (x^{1} x) (x + 8) (x - 8)}{{(x - 8)}^{2} (x + 8)}$

Step 10.4.2

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

$\frac{24 (x^{1} x^{1}) (x + 8) (x - 8)}{{(x - 8)}^{2} (x + 8)}$

Step 10.4.3

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

$\frac{24 x^{1 + 1} (x + 8) (x - 8)}{{(x - 8)}^{2} (x + 8)}$

Step 10.4.4

Add $1$ and $1$ .

$\frac{24 x^{2} (x + 8) (x - 8)}{{(x - 8)}^{2} (x + 8)}$

Step 11

Reduce the expression by cancelling the common factors.

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Step 11.1

Cancel the common factor of $x + 8$ .

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Step 11.1.1

Cancel the common factor.

$\frac{24 x^{2} (x + 8) (x - 8)}{{(x - 8)}^{2} (x + 8)}$

Step 11.1.2

Rewrite the expression.

$\frac{24 x^{2} (x - 8)}{{(x - 8)}^{2}}$

Step 11.2

Cancel the common factor of $x - 8$ and ${(x - 8)}^{2}$ .

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Step 11.2.1

Factor $x - 8$ out of $24 x^{2} (x - 8)$ .

$\frac{(x - 8) (24 x^{2})}{{(x - 8)}^{2}}$

Step 11.2.2

Cancel the common factors.

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Step 11.2.2.1

Factor $x - 8$ out of ${(x - 8)}^{2}$ .

$\frac{(x - 8) (24 x^{2})}{(x - 8) (x - 8)}$

Step 11.2.2.2

Cancel the common factor.

$\frac{(x - 8) (24 x^{2})}{(x - 8) (x - 8)}$

Step 11.2.2.3

Rewrite the expression.

$\frac{24 x^{2}}{x - 8}$