Simplify (x^-5+y^-5)/(x^-4+y^-4)

$\frac{x^{- 5} + y^{- 5}}{x^{- 4} + y^{- 4}}$

Step 1

Simplify the numerator.

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

$\frac{\frac{1}{x^{5}} + y^{- 5}}{x^{- 4} + y^{- 4}}$

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

$\frac{\frac{1}{x^{5}} + \frac{1}{y^{5}}}{x^{- 4} + y^{- 4}}$

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

$\frac{\frac{1}{x^{5}} \cdot \frac{y^{5}}{y^{5}} + \frac{1}{y^{5}}}{x^{- 4} + y^{- 4}}$

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

$\frac{\frac{1}{x^{5}} \cdot \frac{y^{5}}{y^{5}} + \frac{1}{y^{5}} \cdot \frac{x^{5}}{x^{5}}}{x^{- 4} + y^{- 4}}$

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

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

$\frac{\frac{y^{5}}{x^{5} y^{5}} + \frac{1}{y^{5}} \cdot \frac{x^{5}}{x^{5}}}{x^{- 4} + y^{- 4}}$

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

$\frac{\frac{y^{5}}{x^{5} y^{5}} + \frac{x^{5}}{y^{5} x^{5}}}{x^{- 4} + y^{- 4}}$

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

$\frac{\frac{y^{5}}{x^{5} y^{5}} + \frac{x^{5}}{x^{5} y^{5}}}{x^{- 4} + y^{- 4}}$

Combine the numerators over the common denominator.

$\frac{\frac{y^{5} + x^{5}}{x^{5} y^{5}}}{x^{- 4} + y^{- 4}}$

Step 2

Simplify the denominator.

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

$\frac{\frac{y^{5} + x^{5}}{x^{5} y^{5}}}{\frac{1}{x^{4}} + y^{- 4}}$

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

$\frac{\frac{y^{5} + x^{5}}{x^{5} y^{5}}}{\frac{1}{x^{4}} + \frac{1}{y^{4}}}$

To write $\frac{1}{x^{4}}$ as a fraction with a common denominator, multiply by $\frac{y^{4}}{y^{4}}$ .

$\frac{\frac{y^{5} + x^{5}}{x^{5} y^{5}}}{\frac{1}{x^{4}} \cdot \frac{y^{4}}{y^{4}} + \frac{1}{y^{4}}}$

To write $\frac{1}{y^{4}}$ as a fraction with a common denominator, multiply by $\frac{x^{4}}{x^{4}}$ .

$\frac{\frac{y^{5} + x^{5}}{x^{5} y^{5}}}{\frac{1}{x^{4}} \cdot \frac{y^{4}}{y^{4}} + \frac{1}{y^{4}} \cdot \frac{x^{4}}{x^{4}}}$

Write each expression with a common denominator of $x^{4} y^{4}$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{x^{4}}$ by $\frac{y^{4}}{y^{4}}$ .

$\frac{\frac{y^{5} + x^{5}}{x^{5} y^{5}}}{\frac{y^{4}}{x^{4} y^{4}} + \frac{1}{y^{4}} \cdot \frac{x^{4}}{x^{4}}}$

Multiply $\frac{1}{y^{4}}$ by $\frac{x^{4}}{x^{4}}$ .

$\frac{\frac{y^{5} + x^{5}}{x^{5} y^{5}}}{\frac{y^{4}}{x^{4} y^{4}} + \frac{x^{4}}{y^{4} x^{4}}}$

Reorder the factors of $y^{4} x^{4}$ .

$\frac{\frac{y^{5} + x^{5}}{x^{5} y^{5}}}{\frac{y^{4}}{x^{4} y^{4}} + \frac{x^{4}}{x^{4} y^{4}}}$

Combine the numerators over the common denominator.

$\frac{\frac{y^{5} + x^{5}}{x^{5} y^{5}}}{\frac{y^{4} + x^{4}}{x^{4} y^{4}}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{y^{5} + x^{5}}{x^{5} y^{5}} \cdot \frac{x^{4} y^{4}}{y^{4} + x^{4}}$

Step 4

Cancel the common factor of $x^{4} y^{4}$ .

Factor $x^{4} y^{4}$ out of $x^{5} y^{5}$ .

$\frac{y^{5} + x^{5}}{x^{4} y^{4} (x y)} \cdot \frac{x^{4} y^{4}}{y^{4} + x^{4}}$

Cancel the common factor.

$\frac{y^{5} + x^{5}}{x^{4} y^{4} (x y)} \cdot \frac{x^{4} y^{4}}{y^{4} + x^{4}}$

Rewrite the expression.

$\frac{y^{5} + x^{5}}{x y} \cdot \frac{1}{y^{4} + x^{4}}$

Step 5

Multiply $\frac{y^{5} + x^{5}}{x y}$ by $\frac{1}{y^{4} + x^{4}}$ .

$\frac{y^{5} + x^{5}}{x y (y^{4} + x^{4})}$

Simplify (x^-5+y^-5)/(x^-4+y^-4)

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