Simplify (x-1/x)/(5/x+5/(x^2))

$\frac{x - \frac{1}{x}}{\frac{5}{x} + \frac{5}{x^{2}}}$

Step 1

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

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

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

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

Cancel the common factor of $x$ .

Move the leading negative in $- \frac{1}{x}$ into the numerator.

$\frac{x^{2} x + x^{2} \frac{- 1}{x}}{x^{2} \frac{5}{x} + x^{2} \frac{5}{x^{2}}}$

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Simplify the numerator.

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

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

$\frac{x^{2} (x) + x \cdot - 1}{x \cdot 5 + 5}$

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

$\frac{x^{2} (x) + x^{2} (x^{- 1} \cdot - 1)}{x \cdot 5 + 5}$

Factor $x^{2}$ out of $x^{2} (x) + x^{2} (x^{- 1} \cdot - 1)$ .

$\frac{x^{2} (x + x^{- 1} \cdot - 1)}{x \cdot 5 + 5}$

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

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

$\frac{x^{2} (x^{1} + x^{- 1} \cdot - 1)}{x \cdot 5 + 5}$

Factor $x^{- 1}$ out of $x^{1}$ .

$\frac{x^{2} (x^{- 1} x^{2} + x^{- 1} \cdot - 1)}{x \cdot 5 + 5}$

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

$\frac{x^{2} (x^{- 1} x^{2} + x^{- 1} (- 1))}{x \cdot 5 + 5}$

Factor $x^{- 1}$ out of $x^{- 1} x^{2} + x^{- 1} (- 1)$ .

$\frac{x^{2} (x^{- 1} (x^{2} - 1))}{x \cdot 5 + 5}$

Rewrite $1$ as $1^{2}$ .

$\frac{x^{2} (x^{- 1} (x^{2} - 1^{2}))}{x \cdot 5 + 5}$

Factor.

$\frac{x^{2} x^{- 1} (x + 1) (x - 1)}{x \cdot 5 + 5}$

Combine exponents.

Multiply $x^{2}$ by $x^{- 1}$ by adding the exponents.

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

$\frac{x^{2 - 1} (x + 1) (x - 1)}{x \cdot 5 + 5}$

Subtract $1$ from $2$ .

$\frac{x^{1} (x + 1) (x - 1)}{x \cdot 5 + 5}$

Simplify $x^{1}$ .

$\frac{x (x + 1) (x - 1)}{x \cdot 5 + 5}$

Step 5

Simplify terms.

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

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

$\frac{x (x + 1) (x - 1)}{5 \cdot x + 5}$

Factor $5$ out of $5$ .

$\frac{x (x + 1) (x - 1)}{5 \cdot x + 5 (1)}$

Factor $5$ out of $5 \cdot x + 5 (1)$ .

$\frac{x (x + 1) (x - 1)}{5 (x + 1)}$

Cancel the common factor of $x + 1$ .

Cancel the common factor.

$\frac{x (x + 1) (x - 1)}{5 (x + 1)}$

Rewrite the expression.

$\frac{x (x - 1)}{5}$

Simplify (x-1/x)/(5/x+5/(x^2))

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