Algebra Examples

$\frac{\frac{9}{x} - x}{\frac{x^{2}}{3} + x}$

Step 1

Multiply the numerator and denominator of the fraction by $x \cdot 3$ .

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Step 1.1

Multiply $\frac{\frac{9}{x} - x}{\frac{x^{2}}{3} + x}$ by $\frac{x \cdot 3}{x \cdot 3}$ .

$\frac{x \cdot 3}{x \cdot 3} \cdot \frac{\frac{9}{x} - x}{\frac{x^{2}}{3} + x}$

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x$ .

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Step 3.1.1

Factor $x$ out of $x \cdot 3$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Multiply $3$ by $9$ .

$\frac{27 + x \cdot 3 (- x)}{x \cdot 3 \frac{x^{2}}{3} + x \cdot 3 x}$

Step 3.3

Cancel the common factor of $3$ .

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Step 3.3.1

Factor $3$ out of $x \cdot 3$ .

$\frac{27 + x \cdot 3 (- x)}{3 \cdot x \frac{x^{2}}{3} + x \cdot 3 x}$

Step 3.3.2

Cancel the common factor.

$\frac{27 + x \cdot 3 (- x)}{3 \cdot x \frac{x^{2}}{3} + x \cdot 3 x}$

Step 3.3.3

Rewrite the expression.

$\frac{27 + x \cdot 3 (- x)}{x \cdot x^{2} + x \cdot 3 x}$

Step 3.4

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

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Step 3.4.1

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

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Step 3.4.1.1

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

$\frac{27 + x \cdot 3 (- x)}{x^{1} x^{2} + x \cdot 3 x}$

Step 3.4.1.2

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

$\frac{27 + x \cdot 3 (- x)}{x^{1 + 2} + x \cdot 3 x}$

Step 3.4.2

Add $1$ and $2$ .

$\frac{27 + x \cdot 3 (- x)}{x^{3} + x \cdot 3 x}$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $x$ by $x$ by adding the exponents.

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Step 4.1.1

Move $x$ .

$\frac{27 + x \cdot x \cdot 3 \cdot - 1}{x^{3} + x \cdot 3 x}$

Step 4.1.2

Multiply $x$ by $x$ .

$\frac{27 + x^{2} \cdot 3 \cdot - 1}{x^{3} + x \cdot 3 x}$

Step 4.2

Move $3$ to the left of $x^{2}$ .

$\frac{27 + 3 \cdot x^{2} \cdot - 1}{x^{3} + x \cdot 3 x}$

Step 4.3

Multiply $- 1$ by $3$ .

$\frac{27 - 3 x^{2}}{x^{3} + x \cdot 3 x}$

Step 4.4

Rewrite $27 - 3 x^{2}$ in a factored form.

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Step 4.4.1

Factor $3$ out of $27 - 3 x^{2}$ .

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Step 4.4.1.1

Factor $3$ out of $27$ .

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

Step 4.4.1.2

Factor $3$ out of $- 3 x^{2}$ .

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

Step 4.4.1.3

Factor $3$ out of $3 (9) + 3 (- x^{2})$ .

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

Step 4.4.2

Rewrite $9$ as $3^{2}$ .

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

Step 4.4.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = x$ .

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

Step 5

Simplify the denominator.

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Step 5.1

Factor $x$ out of $x^{3} + x \cdot 3 x$ .

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Step 5.1.1

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

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

Step 5.1.2

Factor $x$ out of $x \cdot 3 x$ .

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

Step 5.1.3

Factor $x$ out of $x \cdot x^{2} + x (3 x)$ .

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

Step 5.2

Factor $x$ out of $x^{2} + 3 x$ .

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Step 5.2.1

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

$\frac{3 (3 + x) (3 - x)}{x (x \cdot x + 3 x)}$

Step 5.2.2

Factor $x$ out of $3 x$ .

$\frac{3 (3 + x) (3 - x)}{x (x \cdot x + x \cdot 3)}$

Step 5.2.3

Factor $x$ out of $x \cdot x + x \cdot 3$ .

$\frac{3 (3 + x) (3 - x)}{x (x (x + 3))}$

$\frac{3 (3 + x) (3 - x)}{x \cdot x (x + 3)}$

Step 5.3

Combine exponents.

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Step 5.3.1

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

$\frac{3 (3 + x) (3 - x)}{x^{1} x (x + 3)}$

Step 5.3.2

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

$\frac{3 (3 + x) (3 - x)}{x^{1} x^{1} (x + 3)}$

Step 5.3.3

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

$\frac{3 (3 + x) (3 - x)}{x^{1 + 1} (x + 3)}$

Step 5.3.4

Add $1$ and $1$ .

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

Step 6

Cancel the common factor of $3 + x$ and $x + 3$ .

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Step 6.1

Reorder terms.

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

$\frac{3 (3 - x)}{x^{2}}$