Algebra Examples

$- 4 x y \sqrt{3 x^{2} y^{4}} - 6 x \sqrt[3]{20 x^{2} y^{4}} + 2 x y \sqrt{3 x^{2} y^{4}} + 8 x \sqrt[3]{20 x^{2} y^{4}}$

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

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

$- 4 x y \sqrt{3 x^{2} {(y^{2})}^{2}} - 6 x \sqrt[3]{20 x^{2} y^{4}} + 2 x y \sqrt{3 x^{2} y^{4}} + 8 x \sqrt[3]{20 x^{2} y^{4}}$

Step 1.1.2

Move $3$ .

$- 4 x y \sqrt{x^{2} {(y^{2})}^{2} \cdot 3} - 6 x \sqrt[3]{20 x^{2} y^{4}} + 2 x y \sqrt{3 x^{2} y^{4}} + 8 x \sqrt[3]{20 x^{2} y^{4}}$

Step 1.1.3

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

$- 4 x y \sqrt{{(x y^{2})}^{2} \cdot 3} - 6 x \sqrt[3]{20 x^{2} y^{4}} + 2 x y \sqrt{3 x^{2} y^{4}} + 8 x \sqrt[3]{20 x^{2} y^{4}}$

Step 1.2

Pull terms out from under the radical.

$- 4 x y (x y^{2} \sqrt{3}) - 6 x \sqrt[3]{20 x^{2} y^{4}} + 2 x y \sqrt{3 x^{2} y^{4}} + 8 x \sqrt[3]{20 x^{2} y^{4}}$

Step 1.3

Multiply $x$ by $x$ by adding the exponents.

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Step 1.3.1

Move $x$ .

$- 4 (x \cdot x) y (y^{2} \sqrt{3}) - 6 x \sqrt[3]{20 x^{2} y^{4}} + 2 x y \sqrt{3 x^{2} y^{4}} + 8 x \sqrt[3]{20 x^{2} y^{4}}$

Step 1.3.2

Multiply $x$ by $x$ .

$- 4 x^{2} y (y^{2} \sqrt{3}) - 6 x \sqrt[3]{20 x^{2} y^{4}} + 2 x y \sqrt{3 x^{2} y^{4}} + 8 x \sqrt[3]{20 x^{2} y^{4}}$

Step 1.4

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

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Step 1.4.1

Move $y^{2}$ .

$- 4 x^{2} (y^{2} y) \sqrt{3} - 6 x \sqrt[3]{20 x^{2} y^{4}} + 2 x y \sqrt{3 x^{2} y^{4}} + 8 x \sqrt[3]{20 x^{2} y^{4}}$

Step 1.4.2

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

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Step 1.4.2.1

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

$- 4 x^{2} (y^{2} y^{1}) \sqrt{3} - 6 x \sqrt[3]{20 x^{2} y^{4}} + 2 x y \sqrt{3 x^{2} y^{4}} + 8 x \sqrt[3]{20 x^{2} y^{4}}$

Step 1.4.2.2

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

$- 4 x^{2} y^{2 + 1} \sqrt{3} - 6 x \sqrt[3]{20 x^{2} y^{4}} + 2 x y \sqrt{3 x^{2} y^{4}} + 8 x \sqrt[3]{20 x^{2} y^{4}}$

Step 1.4.3

Add $2$ and $1$ .

$- 4 x^{2} y^{3} \sqrt{3} - 6 x \sqrt[3]{20 x^{2} y^{4}} + 2 x y \sqrt{3 x^{2} y^{4}} + 8 x \sqrt[3]{20 x^{2} y^{4}}$

Step 1.5

Rewrite $20 x^{2} y^{4}$ as $y^{3} \cdot (20 x^{2} y)$ .

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Step 1.5.1

Factor out $y^{3}$ .

$- 4 x^{2} y^{3} \sqrt{3} - 6 x \sqrt[3]{20 x^{2} (y^{3} y)} + 2 x y \sqrt{3 x^{2} y^{4}} + 8 x \sqrt[3]{20 x^{2} y^{4}}$

Step 1.5.2

Move $x^{2}$ .

$- 4 x^{2} y^{3} \sqrt{3} - 6 x \sqrt[3]{20 y^{3} x^{2} y} + 2 x y \sqrt{3 x^{2} y^{4}} + 8 x \sqrt[3]{20 x^{2} y^{4}}$

Step 1.5.3

Reorder $20$ and $y^{3}$ .

$- 4 x^{2} y^{3} \sqrt{3} - 6 x \sqrt[3]{y^{3} \cdot 20 x^{2} y} + 2 x y \sqrt{3 x^{2} y^{4}} + 8 x \sqrt[3]{20 x^{2} y^{4}}$

Step 1.5.4

Add parentheses.

$- 4 x^{2} y^{3} \sqrt{3} - 6 x \sqrt[3]{y^{3} \cdot 20 (x^{2} y)} + 2 x y \sqrt{3 x^{2} y^{4}} + 8 x \sqrt[3]{20 x^{2} y^{4}}$

Step 1.5.5

Add parentheses.

$- 4 x^{2} y^{3} \sqrt{3} - 6 x \sqrt[3]{y^{3} \cdot (20 x^{2} y)} + 2 x y \sqrt{3 x^{2} y^{4}} + 8 x \sqrt[3]{20 x^{2} y^{4}}$

Step 1.6

Pull terms out from under the radical.

$- 4 x^{2} y^{3} \sqrt{3} - 6 x (y \sqrt[3]{20 x^{2} y}) + 2 x y \sqrt{3 x^{2} y^{4}} + 8 x \sqrt[3]{20 x^{2} y^{4}}$

Step 1.7

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

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Step 1.7.1

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

$- 4 x^{2} y^{3} \sqrt{3} - 6 x (y \sqrt[3]{20 x^{2} y}) + 2 x y \sqrt{3 x^{2} {(y^{2})}^{2}} + 8 x \sqrt[3]{20 x^{2} y^{4}}$

Step 1.7.2

Move $3$ .

$- 4 x^{2} y^{3} \sqrt{3} - 6 x (y \sqrt[3]{20 x^{2} y}) + 2 x y \sqrt{x^{2} {(y^{2})}^{2} \cdot 3} + 8 x \sqrt[3]{20 x^{2} y^{4}}$

Step 1.7.3

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

$- 4 x^{2} y^{3} \sqrt{3} - 6 x (y \sqrt[3]{20 x^{2} y}) + 2 x y \sqrt{{(x y^{2})}^{2} \cdot 3} + 8 x \sqrt[3]{20 x^{2} y^{4}}$

Step 1.8

Pull terms out from under the radical.

$- 4 x^{2} y^{3} \sqrt{3} - 6 x (y \sqrt[3]{20 x^{2} y}) + 2 x y (x y^{2} \sqrt{3}) + 8 x \sqrt[3]{20 x^{2} y^{4}}$

Step 1.9

Multiply $x$ by $x$ by adding the exponents.

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Step 1.9.1

Move $x$ .

$- 4 x^{2} y^{3} \sqrt{3} - 6 x (y \sqrt[3]{20 x^{2} y}) + 2 (x \cdot x) y (y^{2} \sqrt{3}) + 8 x \sqrt[3]{20 x^{2} y^{4}}$

Step 1.9.2

Multiply $x$ by $x$ .

$- 4 x^{2} y^{3} \sqrt{3} - 6 x (y \sqrt[3]{20 x^{2} y}) + 2 x^{2} y (y^{2} \sqrt{3}) + 8 x \sqrt[3]{20 x^{2} y^{4}}$

Step 1.10

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

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Step 1.10.1

Move $y^{2}$ .

$- 4 x^{2} y^{3} \sqrt{3} - 6 x (y \sqrt[3]{20 x^{2} y}) + 2 x^{2} (y^{2} y) \sqrt{3} + 8 x \sqrt[3]{20 x^{2} y^{4}}$

Step 1.10.2

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

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Step 1.10.2.1

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

$- 4 x^{2} y^{3} \sqrt{3} - 6 x (y \sqrt[3]{20 x^{2} y}) + 2 x^{2} (y^{2} y^{1}) \sqrt{3} + 8 x \sqrt[3]{20 x^{2} y^{4}}$

Step 1.10.2.2

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

$- 4 x^{2} y^{3} \sqrt{3} - 6 x (y \sqrt[3]{20 x^{2} y}) + 2 x^{2} y^{2 + 1} \sqrt{3} + 8 x \sqrt[3]{20 x^{2} y^{4}}$

Step 1.10.3

Add $2$ and $1$ .

$- 4 x^{2} y^{3} \sqrt{3} - 6 x (y \sqrt[3]{20 x^{2} y}) + 2 x^{2} y^{3} \sqrt{3} + 8 x \sqrt[3]{20 x^{2} y^{4}}$

Step 1.11

Rewrite $20 x^{2} y^{4}$ as $y^{3} \cdot (20 x^{2} y)$ .

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Step 1.11.1

Factor out $y^{3}$ .

$- 4 x^{2} y^{3} \sqrt{3} - 6 x (y \sqrt[3]{20 x^{2} y}) + 2 x^{2} y^{3} \sqrt{3} + 8 x \sqrt[3]{20 x^{2} (y^{3} y)}$

Step 1.11.2

Move $x^{2}$ .

$- 4 x^{2} y^{3} \sqrt{3} - 6 x (y \sqrt[3]{20 x^{2} y}) + 2 x^{2} y^{3} \sqrt{3} + 8 x \sqrt[3]{20 y^{3} x^{2} y}$

Step 1.11.3

Reorder $20$ and $y^{3}$ .

$- 4 x^{2} y^{3} \sqrt{3} - 6 x (y \sqrt[3]{20 x^{2} y}) + 2 x^{2} y^{3} \sqrt{3} + 8 x \sqrt[3]{y^{3} \cdot 20 x^{2} y}$

Step 1.11.4

Add parentheses.

$- 4 x^{2} y^{3} \sqrt{3} - 6 x (y \sqrt[3]{20 x^{2} y}) + 2 x^{2} y^{3} \sqrt{3} + 8 x \sqrt[3]{y^{3} \cdot 20 (x^{2} y)}$

Step 1.11.5

Add parentheses.

$- 4 x^{2} y^{3} \sqrt{3} - 6 x (y \sqrt[3]{20 x^{2} y}) + 2 x^{2} y^{3} \sqrt{3} + 8 x \sqrt[3]{y^{3} \cdot (20 x^{2} y)}$

Step 1.12

Pull terms out from under the radical.

$- 4 x^{2} y^{3} \sqrt{3} - 6 x (y \sqrt[3]{20 x^{2} y}) + 2 x^{2} y^{3} \sqrt{3} + 8 x (y \sqrt[3]{20 x^{2} y})$

Step 2

Simplify by adding terms.

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Step 2.1

Add $- 4 x^{2} y^{3} \sqrt{3}$ and $2 x^{2} y^{3} \sqrt{3}$ .

$- 2 x^{2} y^{3} \sqrt{3} - 6 x (y \sqrt[3]{20 x^{2} y}) + 8 x (y \sqrt[3]{20 x^{2} y})$

Step 2.2

Add $- 6 x (y \sqrt[3]{20 x^{2} y})$ and $8 x (y \sqrt[3]{20 x^{2} y})$ .

$- 2 x^{2} y^{3} \sqrt{3} + 2 x y \sqrt[3]{20 x^{2} y}$