Simplify the Radical Expression square root of x^5y^5*3 square root of 2x^7y^6

$\sqrt{x^{5} y^{5}} \cdot 3 \sqrt{2 x^{7} y^{6}}$

Step 1

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

Factor out $x^{4}$ .

$\sqrt{x^{4} x y^{5}} \cdot 3 \sqrt{2 x^{7} y^{6}}$

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

$\sqrt{{(x^{2})}^{2} x y^{5}} \cdot 3 \sqrt{2 x^{7} y^{6}}$

Factor out $y^{4}$ .

$\sqrt{{(x^{2})}^{2} x (y^{4} y)} \cdot 3 \sqrt{2 x^{7} y^{6}}$

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

$\sqrt{{(x^{2})}^{2} x ({(y^{2})}^{2} y)} \cdot 3 \sqrt{2 x^{7} y^{6}}$

Move $x$ .

$\sqrt{{(x^{2})}^{2} {(y^{2})}^{2} x y} \cdot 3 \sqrt{2 x^{7} y^{6}}$

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

$\sqrt{{(x^{2} y^{2})}^{2} x y} \cdot 3 \sqrt{2 x^{7} y^{6}}$

Add parentheses.

$\sqrt{{(x^{2} y^{2})}^{2} (x y)} \cdot 3 \sqrt{2 x^{7} y^{6}}$

Step 2

Pull terms out from under the radical.

$| x^{2} y^{2} | \sqrt{x y} \cdot 3 \sqrt{2 x^{7} y^{6}}$

Step 3

Remove non-negative terms from the absolute value.

$x^{2} y^{2} \sqrt{x y} \cdot 3 \sqrt{2 x^{7} y^{6}}$

Step 4

Move $3$ to the left of $x^{2} y^{2} \sqrt{x y}$ .

$3 \cdot (x^{2} y^{2} \sqrt{x y}) \sqrt{2 x^{7} y^{6}}$

Step 5

Rewrite $2 x^{7} y^{6}$ as ${(x^{3} y^{3})}^{2} \cdot (2 x)$ .

Factor out $x^{6}$ .

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

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

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

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

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

Move $x$ .

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

Move $2$ .

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

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

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

Add parentheses.

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

Step 6

Pull terms out from under the radical.

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

Step 7

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

Combine using the product rule for radicals.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Step 8

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

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

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

Add parentheses.

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

Step 9

Pull terms out from under the radical.

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

Step 10

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

To multiply absolute values, multiply the terms inside each absolute value.

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

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

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

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

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

Add $1$ and $3$ .

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

Step 11

Remove non-negative terms from the absolute value.

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

Step 12

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

Move $x^{4}$ .

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

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

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

Add $4$ and $2$ .

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

Step 13

Reorder factors in $3 x^{6} y^{2} (| y^{3} | \sqrt{2 y})$ .

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

Simplify the Radical Expression square root of x^5y^5*3 square root of 2x^7y^6

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