Algebra Examples

$\frac{1 + \frac{1}{3 x}}{\frac{3}{2 x} + \frac{1}{2 x^{2}}}$

Step 1

Multiply the numerator and denominator of the fraction by $6 x^{2}$ .

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Step 1.1

Multiply $\frac{1 + \frac{1}{3 x}}{\frac{3}{2 x} + \frac{1}{2 x^{2}}}$ by $\frac{6 x^{2}}{6 x^{2}}$ .

$\frac{6 x^{2}}{6 x^{2}} \cdot \frac{1 + \frac{1}{3 x}}{\frac{3}{2 x} + \frac{1}{2 x^{2}}}$

Step 1.2

Combine.

$\frac{6 x^{2} (1 + \frac{1}{3 x})}{6 x^{2} (\frac{3}{2 x} + \frac{1}{2 x^{2}})}$

Step 2

Apply the distributive property.

$\frac{6 x^{2} \cdot 1 + 6 x^{2} \frac{1}{3 x}}{6 x^{2} \frac{3}{2 x} + 6 x^{2} \frac{1}{2 x^{2}}}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $3 x$ .

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Step 3.1.1

Factor $3 x$ out of $6 x^{2}$ .

$\frac{6 x^{2} \cdot 1 + 3 x (2 x) \frac{1}{3 x}}{6 x^{2} \frac{3}{2 x} + 6 x^{2} \frac{1}{2 x^{2}}}$

Step 3.1.2

Cancel the common factor.

$\frac{6 x^{2} \cdot 1 + 3 x (2 x) \frac{1}{3 x}}{6 x^{2} \frac{3}{2 x} + 6 x^{2} \frac{1}{2 x^{2}}}$

Step 3.1.3

Rewrite the expression.

$\frac{6 x^{2} \cdot 1 + 2 x}{6 x^{2} \frac{3}{2 x} + 6 x^{2} \frac{1}{2 x^{2}}}$

Step 3.2

Cancel the common factor of $2 x$ .

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Step 3.2.1

Factor $2 x$ out of $6 x^{2}$ .

$\frac{6 x^{2} \cdot 1 + 2 x}{2 x (3 x) \frac{3}{2 x} + 6 x^{2} \frac{1}{2 x^{2}}}$

Step 3.2.2

Cancel the common factor.

$\frac{6 x^{2} \cdot 1 + 2 x}{2 x (3 x) \frac{3}{2 x} + 6 x^{2} \frac{1}{2 x^{2}}}$

Step 3.2.3

Rewrite the expression.

$\frac{6 x^{2} \cdot 1 + 2 x}{3 x \cdot 3 + 6 x^{2} \frac{1}{2 x^{2}}}$

Step 3.3

Multiply $3$ by $3$ .

$\frac{6 x^{2} \cdot 1 + 2 x}{9 x + 6 x^{2} \frac{1}{2 x^{2}}}$

Step 3.4

Cancel the common factor of $2 x^{2}$ .

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Step 3.4.1

Factor $2 x^{2}$ out of $6 x^{2}$ .

$\frac{6 x^{2} \cdot 1 + 2 x}{9 x + 2 x^{2} (3) \frac{1}{2 x^{2}}}$

Step 3.4.2

Cancel the common factor.

$\frac{6 x^{2} \cdot 1 + 2 x}{9 x + 2 x^{2} \cdot 3 \frac{1}{2 x^{2}}}$

Step 3.4.3

Rewrite the expression.

$\frac{6 x^{2} \cdot 1 + 2 x}{9 x + 3}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $2 x$ out of $6 x^{2} \cdot 1 + 2 x$ .

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Step 4.1.1

Factor $2 x$ out of $6 x^{2} \cdot 1$ .

$\frac{2 x (3 x \cdot 1) + 2 x}{9 x + 3}$

Step 4.1.2

Factor $2 x$ out of $2 x$ .

$\frac{2 x (3 x \cdot 1) + 2 x (1)}{9 x + 3}$

Step 4.1.3

Factor $2 x$ out of $2 x (3 x \cdot 1) + 2 x (1)$ .

$\frac{2 x (3 x \cdot 1 + 1)}{9 x + 3}$

Step 4.2

Multiply $3$ by $1$ .

$\frac{2 x (3 x + 1)}{9 x + 3}$

Step 5

Simplify terms.

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Step 5.1

Factor $3$ out of $9 x + 3$ .

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Step 5.1.1

Factor $3$ out of $9 x$ .

$\frac{2 x (3 x + 1)}{3 (3 x) + 3}$

Step 5.1.2

Factor $3$ out of $3$ .

$\frac{2 x (3 x + 1)}{3 (3 x) + 3 (1)}$

Step 5.1.3

Factor $3$ out of $3 (3 x) + 3 (1)$ .

$\frac{2 x (3 x + 1)}{3 (3 x + 1)}$

Step 5.2

Cancel the common factor of $3 x + 1$ .

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Step 5.2.1

Cancel the common factor.

$\frac{2 x (3 x + 1)}{3 (3 x + 1)}$

Step 5.2.2

Rewrite the expression.

$\frac{2 x}{3}$