Algebra Examples

$x^{9} + y^{9}$

Step 1

Rewrite $x^{9}$ as ${(x^{3})}^{3}$ .

${(x^{3})}^{3} + y^{9}$

Step 2

Rewrite $y^{9}$ as ${(y^{3})}^{3}$ .

${(x^{3})}^{3} + {(y^{3})}^{3}$

Step 3

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 $a = x^{3}$ and $b = y^{3}$ .

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

Step 4

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 $a = x$ and $b = y$ .

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

Step 5

Factor $- 1$ out of ${(x^{3})}^{2} - x^{3} y^{3} + {(y^{3})}^{2}$ .

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Step 5.1

Factor $- 1$ out of ${(x^{3})}^{2}$ .

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

Step 5.2

Factor $- 1$ out of $- x^{3} y^{3}$ .

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

Step 5.3

Factor $- 1$ out of ${(y^{3})}^{2}$ .

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

Step 5.4

Factor $- 1$ out of $- 1 (- {(x^{3})}^{2}) - (x^{3} y^{3})$ .

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

Step 5.5

Factor $- 1$ out of $- 1 (- {(x^{3})}^{2} + x^{3} y^{3}) - 1 (- {(y^{3})}^{2})$ .

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

Step 5.6

Rewrite $- 1 (- {(x^{3})}^{2} + x^{3} y^{3} - {(y^{3})}^{2})$ as $- (- {(x^{3})}^{2} + x^{3} y^{3} - {(y^{3})}^{2})$ .

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

Step 6

Multiply the exponents in ${(x^{3})}^{2}$ .

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Step 6.1

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

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

Step 6.2

Multiply $3$ by $2$ .

$(x + y) (x^{2} - x y + y^{2}) (- (- x^{6} + x^{3} y^{3} - {(y^{3})}^{2}))$

Step 7

Factor.

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Step 7.1

Multiply the exponents in ${(y^{3})}^{2}$ .

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Step 7.1.1

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

$(x + y) (x^{2} - x y + y^{2}) (- (- x^{6} + x^{3} y^{3} - y^{3 \cdot 2}))$

Step 7.1.2

Multiply $3$ by $2$ .

$(x + y) (x^{2} - x y + y^{2}) (- (- x^{6} + x^{3} y^{3} - y^{6}))$

Step 7.2

Remove unnecessary parentheses.

$(x + y) (x^{2} - x y + y^{2}) \cdot - 1 (- x^{6} + x^{3} y^{3} - y^{6})$

Step 8

Factor out negative.

$- ((x + y) (x^{2} - x y + y^{2}) (- x^{6} + x^{3} y^{3} - y^{6}))$

Step 9

Remove unnecessary parentheses.

$- (x + y) (x^{2} - x y + y^{2}) (- x^{6} + x^{3} y^{3} - y^{6})$

Step 10

Factor.

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Step 10.1

Factor $- 1$ out of $- x^{6} + x^{3} y^{3} - y^{6}$ .

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Step 10.1.1

Factor $- 1$ out of $- x^{6}$ .

$- (x + y) (x^{2} - x y + y^{2}) (- (x^{6}) + x^{3} y^{3} - y^{6})$

Step 10.1.2

Factor $- 1$ out of $x^{3} y^{3}$ .

$- (x + y) (x^{2} - x y + y^{2}) (- (x^{6}) - (- x^{3} y^{3}) - y^{6})$

Step 10.1.3

Factor $- 1$ out of $- y^{6}$ .

$- (x + y) (x^{2} - x y + y^{2}) (- (x^{6}) - (- x^{3} y^{3}) - (y^{6}))$

Step 10.1.4

Factor $- 1$ out of $- (x^{6}) - (- x^{3} y^{3})$ .

$- (x + y) (x^{2} - x y + y^{2}) (- (x^{6} - x^{3} y^{3}) - (y^{6}))$

Step 10.1.5

Factor $- 1$ out of $- (x^{6} - x^{3} y^{3}) - (y^{6})$ .

$- (x + y) (x^{2} - x y + y^{2}) (- (x^{6} - x^{3} y^{3} + y^{6}))$

Step 10.2

Remove unnecessary parentheses.

$- (x + y) (x^{2} - x y + y^{2}) \cdot - 1 (x^{6} - x^{3} y^{3} + y^{6})$

Step 11

Combine exponents.

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Step 11.1

Factor out negative.

$- (((x + y) (x^{2} - x y + y^{2})) \cdot - 1 (x^{6} - x^{3} y^{3} + y^{6}))$

Step 11.2

Multiply $- 1$ by $- 1$ .

$1 (((x + y) (x^{2} - x y + y^{2})) (x^{6} - x^{3} y^{3} + y^{6}))$

Step 11.3

Multiply $x + y$ by $1$ .

$(x + y) ((x^{2} - x y + y^{2}) (x^{6} - x^{3} y^{3} + y^{6}))$

Step 12

Remove unnecessary parentheses.

$(x + y) (x^{2} - x y + y^{2}) (x^{6} - x^{3} y^{3} + y^{6})$