Simplify (1/(5x)+1/(x^2))/((x+7)/10+1/x)

$\frac{\frac{1}{5 x} + \frac{1}{x^{2}}}{\frac{x + 7}{10} + \frac{1}{x}}$

Step 1

Multiply the numerator and denominator of the complex fraction by $10 x^{2}$ .

Multiply $\frac{\frac{1}{5 x} + \frac{1}{x^{2}}}{\frac{x + 7}{10} + \frac{1}{x}}$ by $\frac{10 x^{2}}{10 x^{2}}$ .

$\frac{10 x^{2}}{10 x^{2}} \cdot \frac{\frac{1}{5 x} + \frac{1}{x^{2}}}{\frac{x + 7}{10} + \frac{1}{x}}$

Combine.

$\frac{10 x^{2} (\frac{1}{5 x} + \frac{1}{x^{2}})}{10 x^{2} (\frac{x + 7}{10} + \frac{1}{x})}$

Step 2

Apply the distributive property.

$\frac{10 x^{2} \frac{1}{5 x} + 10 x^{2} \frac{1}{x^{2}}}{10 x^{2} \frac{x + 7}{10} + 10 x^{2} \frac{1}{x}}$

Step 3

Simplify by cancelling.

Cancel the common factor of $5 x$ .

Factor $5 x$ out of $10 x^{2}$ .

$\frac{5 x (2 x) \frac{1}{5 x} + 10 x^{2} \frac{1}{x^{2}}}{10 x^{2} \frac{x + 7}{10} + 10 x^{2} \frac{1}{x}}$

Cancel the common factor.

$\frac{5 x (2 x) \frac{1}{5 x} + 10 x^{2} \frac{1}{x^{2}}}{10 x^{2} \frac{x + 7}{10} + 10 x^{2} \frac{1}{x}}$

Rewrite the expression.

$\frac{2 x + 10 x^{2} \frac{1}{x^{2}}}{10 x^{2} \frac{x + 7}{10} + 10 x^{2} \frac{1}{x}}$

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

Factor $x^{2}$ out of $10 x^{2}$ .

$\frac{2 x + x^{2} \cdot 10 \frac{1}{x^{2}}}{10 x^{2} \frac{x + 7}{10} + 10 x^{2} \frac{1}{x}}$

Cancel the common factor.

$\frac{2 x + x^{2} \cdot 10 \frac{1}{x^{2}}}{10 x^{2} \frac{x + 7}{10} + 10 x^{2} \frac{1}{x}}$

Rewrite the expression.

$\frac{2 x + 10}{10 x^{2} \frac{x + 7}{10} + 10 x^{2} \frac{1}{x}}$

Cancel the common factor of $10$ .

Factor $10$ out of $10 x^{2}$ .

$\frac{2 x + 10}{10 (x^{2}) \frac{x + 7}{10} + 10 x^{2} \frac{1}{x}}$

Cancel the common factor.

$\frac{2 x + 10}{10 x^{2} \frac{x + 7}{10} + 10 x^{2} \frac{1}{x}}$

Rewrite the expression.

$\frac{2 x + 10}{x^{2} (x + 7) + 10 x^{2} \frac{1}{x}}$

Cancel the common factor of $x$ .

Factor $x$ out of $10 x^{2}$ .

$\frac{2 x + 10}{x^{2} (x + 7) + x (10 x) \frac{1}{x}}$

Cancel the common factor.

$\frac{2 x + 10}{x^{2} (x + 7) + x (10 x) \frac{1}{x}}$

Rewrite the expression.

$\frac{2 x + 10}{x^{2} (x + 7) + 10 x}$

Step 4

Factor $2$ out of $2 x + 10$ .

Factor $2$ out of $2 x$ .

$\frac{2 (x) + 10}{x^{2} (x + 7) + 10 x}$

Factor $2$ out of $10$ .

$\frac{2 x + 2 \cdot 5}{x^{2} (x + 7) + 10 x}$

Factor $2$ out of $2 x + 2 \cdot 5$ .

$\frac{2 (x + 5)}{x^{2} (x + 7) + 10 x}$

Step 5

Simplify the denominator.

Factor $x$ out of $x^{2} (x + 7) + 10 x$ .

Factor $x$ out of $x^{2} (x + 7)$ .

$\frac{2 (x + 5)}{x (x (x + 7)) + 10 x}$

Factor $x$ out of $10 x$ .

$\frac{2 (x + 5)}{x (x (x + 7)) + x \cdot 10}$

Factor $x$ out of $x (x (x + 7)) + x \cdot 10$ .

$\frac{2 (x + 5)}{x (x (x + 7) + 10)}$

Apply the distributive property.

$\frac{2 (x + 5)}{x (x \cdot x + x \cdot 7 + 10)}$

Multiply $x$ by $x$ .

$\frac{2 (x + 5)}{x (x^{2} + x \cdot 7 + 10)}$

Move $7$ to the left of $x$ .

$\frac{2 (x + 5)}{x (x^{2} + 7 \cdot x + 10)}$

Factor $x^{2} + 7 x + 10$ using the AC method.

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 $10$ and whose sum is $7$ .

$2, 5$

Write the factored form using these integers.

$\frac{2 (x + 5)}{x ((x + 2) (x + 5))}$

$\frac{2 (x + 5)}{x (x + 2) (x + 5)}$

Step 6

Cancel the common factor of $x + 5$ .

Cancel the common factor.

$\frac{2 (x + 5)}{x (x + 2) (x + 5)}$

Rewrite the expression.

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

Simplify (1/(5x)+1/(x^2))/((x+7)/10+1/x)

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