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

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

Step 1

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

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

Step 2

Simplify the denominator.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Step 3

Simplify terms.

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

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

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

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

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

$\frac{x y^{2}}{y^{2} + x^{2}}$