Simplify square root of x^5y^5*3 square root of 2x^9y^12

$\sqrt{x^{5} y^{5}} \cdot 3 \sqrt{2 x^{9} y^{12}}$

Step 1

Rewrite using the commutative property of multiplication.

$3 \sqrt{x^{5} y^{5}} \sqrt{2 x^{9} y^{12}}$

Step 2

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

Factor out $x^{4}$ .

$3 \sqrt{x^{4} x y^{5}} \sqrt{2 x^{9} y^{12}}$

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

$3 \sqrt{{(x^{2})}^{2} x y^{5}} \sqrt{2 x^{9} y^{12}}$

Factor out $y^{4}$ .

$3 \sqrt{{(x^{2})}^{2} x (y^{4} y)} \sqrt{2 x^{9} y^{12}}$

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

$3 \sqrt{{(x^{2})}^{2} x ({(y^{2})}^{2} y)} \sqrt{2 x^{9} y^{12}}$

Move $x$ .

$3 \sqrt{{(x^{2})}^{2} {(y^{2})}^{2} x y} \sqrt{2 x^{9} y^{12}}$

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

$3 \sqrt{{(x^{2} y^{2})}^{2} x y} \sqrt{2 x^{9} y^{12}}$

Add parentheses.

$3 \sqrt{{(x^{2} y^{2})}^{2} (x y)} \sqrt{2 x^{9} y^{12}}$

Step 3

Pull terms out from under the radical.

$3 (x^{2} y^{2} \sqrt{x y}) \sqrt{2 x^{9} y^{12}}$

Step 4

Rewrite $2 x^{9} y^{12}$ as ${(x^{4} y^{6})}^{2} \cdot (2 x)$ .

Factor out $x^{8}$ .

$3 x^{2} y^{2} \sqrt{x y} \sqrt{2 (x^{8} x) y^{12}}$

Rewrite $x^{8}$ as ${(x^{4})}^{2}$ .

$3 x^{2} y^{2} \sqrt{x y} \sqrt{2 ({(x^{4})}^{2} x) y^{12}}$

Rewrite $y^{12}$ as ${(y^{6})}^{2}$ .

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

Move $x$ .

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

Move $2$ .

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

Rewrite $({(x^{4})}^{2}) {(y^{6})}^{2}$ as ${(x^{4} y^{6})}^{2}$ .

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

Add parentheses.

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

Step 5

Pull terms out from under the radical.

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

Step 6

Multiply $x^{2}$ by $x^{4}$ by adding the exponents.

Move $x^{4}$ .

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

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

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

Add $4$ and $2$ .

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

Step 7

Multiply $y^{2}$ by $y^{6}$ by adding the exponents.

Move $y^{6}$ .

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

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

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

Add $6$ and $2$ .

$3 x^{6} y^{8} \sqrt{x y} \sqrt{2 x}$

Step 8

Multiply $3 x^{6} y^{8} \sqrt{x y} \sqrt{2 x}$ .

Combine using the product rule for radicals.

$3 x^{6} y^{8} \sqrt{2 x (x y)}$

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

$3 x^{6} y^{8} \sqrt{2 (x^{1} x) y}$

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

$3 x^{6} y^{8} \sqrt{2 (x^{1} x^{1}) y}$

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

$3 x^{6} y^{8} \sqrt{2 x^{1 + 1} y}$

Add $1$ and $1$ .

$3 x^{6} y^{8} \sqrt{2 x^{2} y}$

Step 9

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

Reorder $2$ and $x^{2}$ .

$3 x^{6} y^{8} \sqrt{x^{2} \cdot 2 y}$

Add parentheses.

$3 x^{6} y^{8} \sqrt{x^{2} \cdot (2 y)}$

Step 10

Pull terms out from under the radical.

$3 x^{6} y^{8} (x \sqrt{2 y})$

Step 11

Multiply $x^{6}$ by $x$ by adding the exponents.

Move $x$ .

$3 (x \cdot x^{6}) y^{8} \sqrt{2 y}$

Multiply $x$ by $x^{6}$ .

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

$3 (x^{1} x^{6}) y^{8} \sqrt{2 y}$

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

$3 x^{1 + 6} y^{8} \sqrt{2 y}$

Add $1$ and $6$ .

$3 x^{7} y^{8} \sqrt{2 y}$

Simplify square root of x^5y^5*3 square root of 2x^9y^12

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