Algebra Examples

$\frac{x^{2} - 64}{7 x} \cdot \frac{3 x^{3}}{x^{2} + x - 72}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{x^{2} - 8^{2}}{7 x} \cdot \frac{3 x^{3}}{x^{2} + x - 72}$

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 $a = x$ and $b = 8$ .

$\frac{(x + 8) (x - 8)}{7 x} \cdot \frac{3 x^{3}}{x^{2} + x - 72}$

Step 2

Factor $x^{2} + x - 72$ 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 $- 72$ and whose sum is $1$ .

$- 8, 9$

Step 2.2

Write the factored form using these integers.

$\frac{(x + 8) (x - 8)}{7 x} \cdot \frac{3 x^{3}}{(x - 8) (x + 9)}$

Step 3

Simplify terms.

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Step 3.1

Combine.

$\frac{(x + 8) (x - 8) (3 x^{3})}{7 x ((x - 8) (x + 9))}$

Step 3.2

Cancel the common factor of $x - 8$ .

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Step 3.2.1

Cancel the common factor.

$\frac{(x + 8) (x - 8) (3 x^{3})}{7 x ((x - 8) (x + 9))}$

Step 3.2.2

Rewrite the expression.

$\frac{(x + 8) (3 x^{3})}{7 x (x + 9)}$

Step 3.3

Cancel the common factor of $x^{3}$ and $x$ .

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Step 3.3.1

Factor $x$ out of $(x + 8) (3 x^{3})$ .

$\frac{x ((x + 8) (3 x^{2}))}{7 x (x + 9)}$

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

Factor $x$ out of $7 x (x + 9)$ .

$\frac{x ((x + 8) (3 x^{2}))}{x (7 (x + 9))}$

Step 3.3.2.2

Cancel the common factor.

$\frac{x ((x + 8) (3 x^{2}))}{x (7 (x + 9))}$

Step 3.3.2.3

Rewrite the expression.

$\frac{(x + 8) (3 x^{2})}{7 (x + 9)}$

Step 3.4

Simplify the expression.

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Step 3.4.1

Move $3$ to the left of $x + 8$ .

$\frac{3 \cdot (x + 8) x^{2}}{7 (x + 9)}$

Step 3.4.2

Reorder factors in $\frac{3 (x + 8) x^{2}}{7 (x + 9)}$ .

$\frac{3 x^{2} (x + 8)}{7 (x + 9)}$