Simplify (1+1/(3x))/(3/(2x)+1/(2x^2))

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

Step 1

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

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

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

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.

Cancel the common factor of $3 x$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $2 x$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $3$ by $3$ .

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Simplify the numerator.

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

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

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

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

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

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

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

Multiply $3$ by $1$ .

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

Step 5

Simplify terms.

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

Factor $3$ out of $9 x$ .

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

Factor $3$ out of $3$ .

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{2 x}{3}$

Simplify (1+1/(3x))/(3/(2x)+1/(2x^2))

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