Algebra Examples

$\sqrt{3 x^{2} y^{2}} \cdot \sqrt{12 x^{5}}$

Step 1

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

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Step 1.1

Move $3$ .

$\sqrt{x^{2} y^{2} \cdot 3} \cdot \sqrt{12 x^{5}}$

Step 1.2

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

$\sqrt{{(x y)}^{2} \cdot 3} \cdot \sqrt{12 x^{5}}$

Step 2

Pull terms out from under the radical.

$x y \sqrt{3} \cdot \sqrt{12 x^{5}}$

Step 3

Rewrite $12 x^{5}$ as ${(2 x^{2})}^{2} \cdot (3 x)$ .

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Step 3.1

Factor $4$ out of $12$ .

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

Step 3.2

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

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

Step 3.3

Factor out $x^{4}$ .

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

Step 3.4

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

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

Step 3.5

Move $3$ .

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

Step 3.6

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

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

Step 3.7

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

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Step 5.1

Move $x^{2}$ .

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.3

Add $2$ and $1$ .

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

Step 6

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

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Step 6.1

Combine using the product rule for radicals.

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

Step 6.2

Multiply $3$ by $3$ .

$x^{3} y (2 \sqrt{9 x})$

Step 7

Simplify the expression.

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Step 7.1

Rewrite using the commutative property of multiplication.

$2 x^{3} y \sqrt{9 x}$

Step 7.2

Rewrite $9$ as $3^{2}$ .

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

Step 8

Pull terms out from under the radical.

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

Step 9

Multiply $3$ by $2$ .

$6 x^{3} y \sqrt{x}$