Algebra Examples

$\sqrt{4 x^{3} y^{3}} \cdot \sqrt{5 x^{3} y^{5}}$

Step 1

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

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Step 1.1

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

$\sqrt{2^{2} x^{3} y^{3}} \cdot \sqrt{5 x^{3} y^{5}}$

Step 1.2

Factor out $x^{2}$ .

$\sqrt{2^{2} (x^{2} x) y^{3}} \cdot \sqrt{5 x^{3} y^{5}}$

Step 1.3

Factor out $y^{2}$ .

$\sqrt{2^{2} (x^{2} x) (y^{2} y)} \cdot \sqrt{5 x^{3} y^{5}}$

Step 1.4

Move $x$ .

$\sqrt{2^{2} (x^{2}) y^{2} x y} \cdot \sqrt{5 x^{3} y^{5}}$

Step 1.5

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

$\sqrt{{(2 x y)}^{2} x y} \cdot \sqrt{5 x^{3} y^{5}}$

Step 1.6

Add parentheses.

$\sqrt{{(2 x y)}^{2} (x y)} \cdot \sqrt{5 x^{3} y^{5}}$

Step 2

Pull terms out from under the radical.

$2 x y \sqrt{x y} \cdot \sqrt{5 x^{3} y^{5}}$

Step 3

Rewrite $5 x^{3} y^{5}$ as ${(x y^{2})}^{2} \cdot (5 x y)$ .

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Step 3.1

Factor out $x^{2}$ .

$2 x y \sqrt{x y} \cdot \sqrt{5 (x^{2} x) y^{5}}$

Step 3.2

Factor out $y^{4}$ .

$2 x y \sqrt{x y} \cdot \sqrt{5 (x^{2} x) (y^{4} y)}$

Step 3.3

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

$2 x y \sqrt{x y} \cdot \sqrt{5 (x^{2} x) ({(y^{2})}^{2} y)}$

Step 3.4

Move $x$ .

$2 x y \sqrt{x y} \cdot \sqrt{5 (x^{2}) {(y^{2})}^{2} x y}$

Step 3.5

Move $5$ .

$2 x y \sqrt{x y} \cdot \sqrt{x^{2} {(y^{2})}^{2} \cdot 5 x y}$

Step 3.6

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

$2 x y \sqrt{x y} \cdot \sqrt{{(x y^{2})}^{2} \cdot 5 x y}$

Step 3.7

Add parentheses.

$2 x y \sqrt{x y} \cdot \sqrt{{(x y^{2})}^{2} \cdot 5 (x y)}$

Step 3.8

Add parentheses.

$2 x y \sqrt{x y} \cdot \sqrt{{(x y^{2})}^{2} \cdot (5 x y)}$

Step 4

Pull terms out from under the radical.

$2 x y \sqrt{x y} \cdot (x y^{2} \sqrt{5 x y})$

Step 5

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

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Step 5.1

Move $x$ .

$2 (x \cdot x) y \sqrt{x y} \cdot (y^{2} \sqrt{5 x y})$

Step 5.2

Multiply $x$ by $x$ .

$2 x^{2} y \sqrt{x y} \cdot (y^{2} \sqrt{5 x y})$

Step 6

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

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Step 6.1

Move $y^{2}$ .

$2 x^{2} (y^{2} y) \sqrt{x y} \cdot \sqrt{5 x y}$

Step 6.2

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

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Step 6.2.1

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

$2 x^{2} (y^{2} y^{1}) \sqrt{x y} \cdot \sqrt{5 x y}$

Step 6.2.2

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

$2 x^{2} y^{2 + 1} \sqrt{x y} \cdot \sqrt{5 x y}$

Step 6.3

Add $2$ and $1$ .

$2 x^{2} y^{3} \sqrt{x y} \cdot \sqrt{5 x y}$

Step 7

Multiply $2 x^{2} y^{3} \sqrt{x y} \sqrt{5 x y}$ .

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Step 7.1

Combine using the product rule for radicals.

$2 x^{2} y^{3} \sqrt{5 x y (x y)}$

Step 7.2

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

$2 x^{2} y^{3} \sqrt{5 (x^{1} x) y \cdot y}$

Step 7.3

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

$2 x^{2} y^{3} \sqrt{5 (x^{1} x^{1}) y \cdot y}$

Step 7.4

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

$2 x^{2} y^{3} \sqrt{5 x^{1 + 1} y \cdot y}$

Step 7.5

Add $1$ and $1$ .

$2 x^{2} y^{3} \sqrt{5 x^{2} y \cdot y}$

Step 7.6

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

$2 x^{2} y^{3} \sqrt{5 x^{2} (y^{1} y)}$

Step 7.7

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

$2 x^{2} y^{3} \sqrt{5 x^{2} (y^{1} y^{1})}$

Step 7.8

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

$2 x^{2} y^{3} \sqrt{5 x^{2} y^{1 + 1}}$

Step 7.9

Add $1$ and $1$ .

$2 x^{2} y^{3} \sqrt{5 x^{2} y^{2}}$

Step 8

Rewrite $5 x^{2} y^{2}$ as ${(x y)}^{2} \cdot 5$ .

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Step 8.1

Move $5$ .

$2 x^{2} y^{3} \sqrt{x^{2} y^{2} \cdot 5}$

Step 8.2

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

$2 x^{2} y^{3} \sqrt{{(x y)}^{2} \cdot 5}$

Step 9

Pull terms out from under the radical.

$2 x^{2} y^{3} (x y \sqrt{5})$

Step 10

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

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Step 10.1

Move $x$ .

$2 (x \cdot x^{2}) y^{3} (y \sqrt{5})$

Step 10.2

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

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Step 10.2.1

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

$2 (x^{1} x^{2}) y^{3} (y \sqrt{5})$

Step 10.2.2

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

$2 x^{1 + 2} y^{3} (y \sqrt{5})$

Step 10.3

Add $1$ and $2$ .

$2 x^{3} y^{3} (y \sqrt{5})$

Step 11

Multiply $y^{3}$ by $y$ by adding the exponents.

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Step 11.1

Move $y$ .

$2 x^{3} (y \cdot y^{3}) \sqrt{5}$

Step 11.2

Multiply $y$ by $y^{3}$ .

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Step 11.2.1

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

$2 x^{3} (y^{1} y^{3}) \sqrt{5}$

Step 11.2.2

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

$2 x^{3} y^{1 + 3} \sqrt{5}$

Step 11.3

Add $1$ and $3$ .

$2 x^{3} y^{4} \sqrt{5}$