Finite Math Examples

$\sqrt{12 x^{5} y} \cdot \sqrt{3 x^{6} y}$

Step 1

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

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Step 1.1

Factor $4$ out of $12$ .

$\sqrt{4 (3) x^{5} y} \cdot \sqrt{3 x^{6} y}$

Step 1.2

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

$\sqrt{2^{2} \cdot 3 x^{5} y} \cdot \sqrt{3 x^{6} y}$

Step 1.3

Factor out $x^{4}$ .

$\sqrt{2^{2} \cdot 3 (x^{4} x) y} \cdot \sqrt{3 x^{6} y}$

Step 1.4

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

$\sqrt{2^{2} \cdot 3 ({(x^{2})}^{2} x) y} \cdot \sqrt{3 x^{6} y}$

Step 1.5

Move $3$ .

$\sqrt{2^{2} {(x^{2})}^{2} \cdot 3 x y} \cdot \sqrt{3 x^{6} y}$

Step 1.6

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

$\sqrt{{(2 x^{2})}^{2} \cdot 3 x y} \cdot \sqrt{3 x^{6} y}$

Step 1.7

Add parentheses.

$\sqrt{{(2 x^{2})}^{2} \cdot 3 (x y)} \cdot \sqrt{3 x^{6} y}$

Step 1.8

Add parentheses.

$\sqrt{{(2 x^{2})}^{2} \cdot (3 x y)} \cdot \sqrt{3 x^{6} y}$

Step 2

Pull terms out from under the radical.

$2 x^{2} \sqrt{3 x y} \cdot \sqrt{3 x^{6} y}$

Step 3

Rewrite $3 x^{6} y$ as ${(x^{3})}^{2} \cdot (3 y)$ .

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Step 3.1

Rewrite $x^{6}$ as ${(x^{3})}^{2}$ .

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

Step 3.2

Reorder $3$ and ${(x^{3})}^{2}$ .

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

Step 3.3

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 5.1

Move $x^{3}$ .

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

Step 5.2

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

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

Step 5.3

Add $3$ and $2$ .

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

Step 6

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

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Step 6.1

Combine using the product rule for radicals.

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

Step 6.2

Multiply $3$ by $3$ .

$2 x^{5} \sqrt{9 y (x y)}$

Step 6.3

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

$2 x^{5} \sqrt{9 (x (y^{1} y))}$

Step 6.4

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

$2 x^{5} \sqrt{9 (x (y^{1} y^{1}))}$

Step 6.5

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

$2 x^{5} \sqrt{9 (x y^{1 + 1})}$

Step 6.6

Add $1$ and $1$ .

$2 x^{5} \sqrt{9 (x y^{2})}$

Step 7

Rewrite $9 x y^{2}$ as ${(3 y)}^{2} x$ .

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Step 7.1

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

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

Step 7.2

Move $x$ .

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

Step 7.3

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

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

Step 8

Pull terms out from under the radical.

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

Step 9

Simplify the expression.

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Step 9.1

Multiply $3$ by $2$ .

$6 x^{5} (y \sqrt{x})$

Step 9.2

Reorder factors in $6 x^{5} (y \sqrt{x})$ .

$6 x^{5} y \sqrt{x}$