Basic Math Examples

\sqrt{y^{5} - y^{2}} + \sqrt{81 y^{3} - 81}

$\sqrt{y^{5} - y^{2}} + \sqrt{81 y^{3} - 81}$

Step 1

Factor

$y^{2}$ out of

$y^{5} - y^{2}$ .

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Step 1.1

Factor

$y^{2}$ out of

$y^{5}$ .

$\sqrt{y^{2} y^{3} - y^{2}} + \sqrt{81 y^{3} - 81}$

Step 1.2

Factor

$y^{2}$ out of

$- y^{2}$ .

$\sqrt{y^{2} y^{3} + y^{2} \cdot - 1} + \sqrt{81 y^{3} - 81}$

Step 1.3

Factor

$y^{2}$ out of

$y^{2} y^{3} + y^{2} \cdot - 1$ .

$\sqrt{y^{2} (y^{3} - 1)} + \sqrt{81 y^{3} - 81}$

Step 2

Rewrite

$1$ as

$1^{3}$ .

$\sqrt{y^{2} (y^{3} - 1^{3})} + \sqrt{81 y^{3} - 81}$

Step 3

Since both terms are perfect cubes, factor using the difference of cubes formula,

$a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where

$a = y$ and

$b = 1$ .

$\sqrt{y^{2} ((y - 1) (y^{2} + y \cdot 1 + 1^{2}))} + \sqrt{81 y^{3} - 81}$

Step 4

Factor.

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Step 4.1

Simplify.

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Step 4.1.1

Multiply

$y$ by

$1$ .

$\sqrt{y^{2} ((y - 1) (y^{2} + y + 1^{2}))} + \sqrt{81 y^{3} - 81}$

Step 4.1.2

One to any power is one.

$\sqrt{y^{2} ((y - 1) (y^{2} + y + 1))} + \sqrt{81 y^{3} - 81}$

Step 4.2

Remove unnecessary parentheses.

$\sqrt{y^{2} (y - 1) (y^{2} + y + 1)} + \sqrt{81 y^{3} - 81}$

Step 5

Rewrite

$y^{2} (y - 1) (y^{2} + y + 1)$ as

$y^{2} ((y - 1) (y^{2} + y + 1^{2}))$ .

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Step 5.1

Rewrite

$1$ as

$1^{2}$ .

$\sqrt{y^{2} (y - 1) (y^{2} + y + 1^{2})} + \sqrt{81 y^{3} - 81}$

Step 5.2

Add parentheses.

$\sqrt{y^{2} ((y - 1) (y^{2} + y + 1^{2}))} + \sqrt{81 y^{3} - 81}$

Step 6

Pull terms out from under the radical.

$y \sqrt{(y - 1) (y^{2} + y + 1^{2})} + \sqrt{81 y^{3} - 81}$

Step 7

One to any power is one.

$y \sqrt{(y - 1) (y^{2} + y + 1)} + \sqrt{81 y^{3} - 81}$

Step 8

Factor

$81$ out of

$81 y^{3} - 81$ .

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Step 8.1

Factor

$81$ out of

$81 y^{3}$ .

$y \sqrt{(y - 1) (y^{2} + y + 1)} + \sqrt{81 (y^{3}) - 81}$

Step 8.2

Factor

$81$ out of

$- 81$ .

$y \sqrt{(y - 1) (y^{2} + y + 1)} + \sqrt{81 (y^{3}) + 81 (- 1)}$

Step 8.3

Factor

$81$ out of

$81 (y^{3}) + 81 (- 1)$ .

$y \sqrt{(y - 1) (y^{2} + y + 1)} + \sqrt{81 (y^{3} - 1)}$

Step 9

Rewrite

$1$ as

$1^{3}$ .

$y \sqrt{(y - 1) (y^{2} + y + 1)} + \sqrt{81 (y^{3} - 1^{3})}$

Step 10

Since both terms are perfect cubes, factor using the difference of cubes formula,

$a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where

$a = y$ and

$b = 1$ .

$y \sqrt{(y - 1) (y^{2} + y + 1)} + \sqrt{81 ((y - 1) (y^{2} + y \cdot 1 + 1^{2}))}$

Step 11

Factor.

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Step 11.1

Simplify.

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Step 11.1.1

Multiply

$y$ by

$1$ .

$y \sqrt{(y - 1) (y^{2} + y + 1)} + \sqrt{81 ((y - 1) (y^{2} + y + 1^{2}))}$

Step 11.1.2

One to any power is one.

$y \sqrt{(y - 1) (y^{2} + y + 1)} + \sqrt{81 ((y - 1) (y^{2} + y + 1))}$

Step 11.2

Remove unnecessary parentheses.

$y \sqrt{(y - 1) (y^{2} + y + 1)} + \sqrt{81 (y - 1) (y^{2} + y + 1)}$

Step 12

Rewrite

$81 (y - 1) (y^{2} + y + 1)$ as

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

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Step 12.1

Rewrite

$81$ as

$9^{2}$ .

$y \sqrt{(y - 1) (y^{2} + y + 1)} + \sqrt{9^{2} (y - 1) (y^{2} + y + 1)}$

Step 12.2

Rewrite

$9$ as

$3^{2}$ .

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

Step 12.3

Add parentheses.

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

Step 13

Pull terms out from under the radical.

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

Step 14

Raise

$3$ to the power of

$2$ .

$y \sqrt{(y - 1) (y^{2} + y + 1)} + 9 \sqrt{(y - 1) (y^{2} + y + 1)}$

Step 15

Factor

$\sqrt{(y - 1) (y^{2} + y + 1)}$ out of

$y \sqrt{(y - 1) (y^{2} + y + 1)} + 9 \sqrt{(y - 1) (y^{2} + y + 1)}$ .

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Step 15.1

Factor

$\sqrt{(y - 1) (y^{2} + y + 1)}$ out of

$y \sqrt{(y - 1) (y^{2} + y + 1)}$ .

$\sqrt{(y - 1) (y^{2} + y + 1)} y + 9 \sqrt{(y - 1) (y^{2} + y + 1)}$

Step 15.2

Factor

$\sqrt{(y - 1) (y^{2} + y + 1)}$ out of

$9 \sqrt{(y - 1) (y^{2} + y + 1)}$ .

$\sqrt{(y - 1) (y^{2} + y + 1)} y + \sqrt{(y - 1) (y^{2} + y + 1)} \cdot 9$

Step 15.3

Factor

$\sqrt{(y - 1) (y^{2} + y + 1)}$ out of

$\sqrt{(y - 1) (y^{2} + y + 1)} y + \sqrt{(y - 1) (y^{2} + y + 1)} \cdot 9$ .

$\sqrt{(y - 1) (y^{2} + y + 1)} (y + 9)$