Algebra Examples

$\sqrt{8 {(x + y)}^{3}} - \sqrt{32 {(x + y)}^{3}}$

Step 1

Simplify each term.

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Step 1.1

Rewrite $8 {(x + y)}^{3}$ as ${(2 (x + y))}^{2} \cdot (2 (x + y))$ .

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Step 1.1.1

Factor $4$ out of $8$ .

$\sqrt{4 (2) {(x + y)}^{3}} - \sqrt{32 {(x + y)}^{3}}$

Step 1.1.2

Rewrite $4$ as $2^{2}$ .

$\sqrt{2^{2} \cdot 2 {(x + y)}^{3}} - \sqrt{32 {(x + y)}^{3}}$

Step 1.1.3

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

$\sqrt{2^{2} \cdot 2 ({(x + y)}^{2} (x + y))} - \sqrt{32 {(x + y)}^{3}}$

Step 1.1.4

Move $2$ .

$\sqrt{2^{2} {(x + y)}^{2} \cdot 2 (x + y)} - \sqrt{32 {(x + y)}^{3}}$

Step 1.1.5

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

$\sqrt{{(2 (x + y))}^{2} \cdot 2 (x + y)} - \sqrt{32 {(x + y)}^{3}}$

Step 1.1.6

Add parentheses.

$\sqrt{{(2 (x + y))}^{2} \cdot (2 (x + y))} - \sqrt{32 {(x + y)}^{3}}$

Step 1.2

Pull terms out from under the radical.

$2 (x + y) \sqrt{2 (x + y)} - \sqrt{32 {(x + y)}^{3}}$

Step 1.3

Apply the distributive property.

$(2 x + 2 y) \sqrt{2 (x + y)} - \sqrt{32 {(x + y)}^{3}}$

Step 1.4

Apply the distributive property.

$2 x \sqrt{2 (x + y)} + 2 y \sqrt{2 (x + y)} - \sqrt{32 {(x + y)}^{3}}$

Step 1.5

Rewrite $32 {(x + y)}^{3}$ as ${(4 (x + y))}^{2} \cdot (2 (x + y))$ .

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Step 1.5.1

Factor $16$ out of $32$ .

$2 x \sqrt{2 (x + y)} + 2 y \sqrt{2 (x + y)} - \sqrt{16 (2) {(x + y)}^{3}}$

Step 1.5.2

Rewrite $16$ as $4^{2}$ .

$2 x \sqrt{2 (x + y)} + 2 y \sqrt{2 (x + y)} - \sqrt{4^{2} \cdot 2 {(x + y)}^{3}}$

Step 1.5.3

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

$2 x \sqrt{2 (x + y)} + 2 y \sqrt{2 (x + y)} - \sqrt{4^{2} \cdot 2 ({(x + y)}^{2} (x + y))}$

Step 1.5.4

Move $2$ .

$2 x \sqrt{2 (x + y)} + 2 y \sqrt{2 (x + y)} - \sqrt{4^{2} {(x + y)}^{2} \cdot 2 (x + y)}$

Step 1.5.5

Rewrite $4^{2} {(x + y)}^{2}$ as ${(4 (x + y))}^{2}$ .

$2 x \sqrt{2 (x + y)} + 2 y \sqrt{2 (x + y)} - \sqrt{{(4 (x + y))}^{2} \cdot 2 (x + y)}$

Step 1.5.6

Add parentheses.

$2 x \sqrt{2 (x + y)} + 2 y \sqrt{2 (x + y)} - \sqrt{{(4 (x + y))}^{2} \cdot (2 (x + y))}$

Step 1.6

Pull terms out from under the radical.

$2 x \sqrt{2 (x + y)} + 2 y \sqrt{2 (x + y)} - (4 (x + y) \sqrt{2 (x + y)})$

Step 1.7

Apply the distributive property.

$2 x \sqrt{2 (x + y)} + 2 y \sqrt{2 (x + y)} - ((4 x + 4 y) \sqrt{2 (x + y)})$

Step 1.8

Apply the distributive property.

$2 x \sqrt{2 (x + y)} + 2 y \sqrt{2 (x + y)} - (4 x \sqrt{2 (x + y)} + 4 y \sqrt{2 (x + y)})$

Step 1.9

Apply the distributive property.

$2 x \sqrt{2 (x + y)} + 2 y \sqrt{2 (x + y)} - (4 x \sqrt{2 (x + y)}) - (4 y \sqrt{2 (x + y)})$

Step 1.10

Multiply $4$ by $- 1$ .

$2 x \sqrt{2 (x + y)} + 2 y \sqrt{2 (x + y)} - 4 (x \sqrt{2 (x + y)}) - (4 y \sqrt{2 (x + y)})$

Step 1.11

Multiply $4$ by $- 1$ .

$2 x \sqrt{2 (x + y)} + 2 y \sqrt{2 (x + y)} - 4 (x \sqrt{2 (x + y)}) - 4 (y \sqrt{2 (x + y)})$

Step 1.12

Remove parentheses.

$2 x \sqrt{2 (x + y)} + 2 y \sqrt{2 (x + y)} - 4 x \sqrt{2 (x + y)} - 4 y \sqrt{2 (x + y)}$

Step 2

Simplify by adding terms.

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Step 2.1

Subtract $4 x \sqrt{2 (x + y)}$ from $2 x \sqrt{2 (x + y)}$ .

$- 2 x \sqrt{2 (x + y)} + 2 y \sqrt{2 (x + y)} - 4 y \sqrt{2 (x + y)}$

Step 2.2

Subtract $4 y \sqrt{2 (x + y)}$ from $2 y \sqrt{2 (x + y)}$ .

$- 2 x \sqrt{2 (x + y)} - 2 y \sqrt{2 (x + y)}$