Simplify (x^-3+y^-3)/(x^-2+y^-2)

$\frac{x^{- 3} + y^{- 3}}{x^{- 2} + y^{- 2}}$

Step 1

Simplify the numerator.

Rewrite $x^{- 3}$ as ${(x^{- 1})}^{3}$ .

$\frac{{(x^{- 1})}^{3} + y^{- 3}}{x^{- 2} + y^{- 2}}$

Rewrite $y^{- 3}$ as ${(y^{- 1})}^{3}$ .

$\frac{{(x^{- 1})}^{3} + {(y^{- 1})}^{3}}{x^{- 2} + y^{- 2}}$

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x^{- 1}$ and $b = y^{- 1}$ .

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

Simplify.

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

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

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

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

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

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

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

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

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

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

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

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

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

Reorder the factors of $y x$ .

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

Combine the numerators over the common denominator.

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

Multiply the exponents in ${(x^{- 1})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Multiply $- 1$ by $2$ .

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

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

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

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

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

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

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

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

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

Multiply the exponents in ${(y^{- 1})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Multiply $- 1$ by $2$ .

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Combine the numerators over the common denominator.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

Combine the numerators over the common denominator.

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

Rewrite $\frac{(y - x) y + x^{2}}{x^{2} y^{2}}$ in a factored form.

Apply the distributive property.

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

Multiply $y$ by $y$ .

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

Step 2

Simplify the denominator.

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

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

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

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

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

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

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

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

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 $\frac{y^{2}}{y^{2}}$ .

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

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

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

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

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

Combine the numerators over the common denominator.

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

Step 3

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

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

Step 4

Simplify the denominator.

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

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

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

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

Add $2$ and $1$ .

$\frac{\frac{(y + x) (y^{2} - x y + x^{2})}{x^{3} y \cdot y^{2}}}{\frac{y^{2} + x^{2}}{x^{2} y^{2}}}$

Multiply $y$ by $y^{2}$ by adding the exponents.

Move $y^{2}$ .

$\frac{\frac{(y + x) (y^{2} - x y + x^{2})}{x^{3} (y^{2} y)}}{\frac{y^{2} + x^{2}}{x^{2} y^{2}}}$

Multiply $y^{2}$ by $y$ .

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

$\frac{\frac{(y + x) (y^{2} - x y + x^{2})}{x^{3} (y^{2} y^{1})}}{\frac{y^{2} + x^{2}}{x^{2} y^{2}}}$

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

$\frac{\frac{(y + x) (y^{2} - x y + x^{2})}{x^{3} y^{2 + 1}}}{\frac{y^{2} + x^{2}}{x^{2} y^{2}}}$

Add $2$ and $1$ .

$\frac{\frac{(y + x) (y^{2} - x y + x^{2})}{x^{3} y^{3}}}{\frac{y^{2} + x^{2}}{x^{2} y^{2}}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\frac{(y + x) (y^{2} - x y + x^{2})}{x^{3} y^{3}} \cdot \frac{x^{2} y^{2}}{y^{2} + x^{2}}$

Step 6

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

Factor $x^{2} y^{2}$ out of $x^{3} y^{3}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 7

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

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

Simplify (x^-3+y^-3)/(x^-2+y^-2)

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