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

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

Step 1

Move $x^{- 1}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

Simplify the denominator.

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

To write $\frac{1}{x^{2}}$ as a fraction with a common denominator, multiply by $\frac{y^{5}}{y^{5}}$ .

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

To write $- \frac{1}{y^{5}}$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

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

Write each expression with a common denominator of $x^{2} y^{5}$ , by multiplying each by an appropriate factor of $1$ .

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

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

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

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

Reorder the factors of $x^{2} y^{5}$ .

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

Combine the numerators over the common denominator.

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

Step 3

Simplify terms.

Combine $\frac{y^{5} - x^{2}}{y^{5} x^{2}}$ and $x$ .

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

Reduce the expression $\frac{(y^{5} - x^{2}) x}{y^{5} x^{2}}$ by cancelling the common factors.

Factor $x$ out of $(y^{5} - x^{2}) x$ .

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

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

$\frac{y^{5} x}{y^{5} - x^{2}}$