Simplify square root of 54x^4y^3* square root of 72xy^6

$\sqrt{54 x^{4} y^{3}} \cdot \sqrt{72 x y^{6}}$

Step 1

Rewrite $54 x^{4} y^{3}$ as ${(3 x^{2} y)}^{2} \cdot (6 y)$ .

Factor $9$ out of $54$ .

$\sqrt{9 (6) x^{4} y^{3}} \cdot \sqrt{72 x y^{6}}$

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

$\sqrt{3^{2} \cdot 6 x^{4} y^{3}} \cdot \sqrt{72 x y^{6}}$

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

$\sqrt{3^{2} \cdot 6 {(x^{2})}^{2} y^{3}} \cdot \sqrt{72 x y^{6}}$

Factor out $y^{2}$ .

$\sqrt{3^{2} \cdot 6 {(x^{2})}^{2} (y^{2} y)} \cdot \sqrt{72 x y^{6}}$

Move $6$ .

$\sqrt{3^{2} {(x^{2})}^{2} y^{2} \cdot 6 y} \cdot \sqrt{72 x y^{6}}$

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

$\sqrt{{(3 x^{2} y)}^{2} \cdot 6 y} \cdot \sqrt{72 x y^{6}}$

Add parentheses.

$\sqrt{{(3 x^{2} y)}^{2} \cdot (6 y)} \cdot \sqrt{72 x y^{6}}$

Step 2

Pull terms out from under the radical.

$3 x^{2} y \sqrt{6 y} \cdot \sqrt{72 x y^{6}}$

Step 3

Rewrite $72 x y^{6}$ as ${(6 y^{3})}^{2} \cdot (2 x)$ .

Factor $36$ out of $72$ .

$3 x^{2} y \sqrt{6 y} \cdot \sqrt{36 (2) x y^{6}}$

Rewrite $36$ as $6^{2}$ .

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

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

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

Move $x$ .

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

Move $2$ .

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

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

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

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

Move $y^{3}$ .

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

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

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

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

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

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

Add $3$ and $1$ .

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

Step 6

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

Multiply $6$ by $3$ .

$18 x^{2} y^{4} \sqrt{6 y} \sqrt{2 x}$

Combine using the product rule for radicals.

$18 x^{2} y^{4} \sqrt{2 x (6 y)}$

Multiply $6$ by $2$ .

$18 x^{2} y^{4} \sqrt{12 x y}$

Step 7

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

Factor $4$ out of $12$ .

$18 x^{2} y^{4} \sqrt{4 (3) x y}$

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

$18 x^{2} y^{4} \sqrt{2^{2} \cdot 3 x y}$

Add parentheses.

$18 x^{2} y^{4} \sqrt{2^{2} \cdot 3 (x y)}$

Add parentheses.

$18 x^{2} y^{4} \sqrt{2^{2} \cdot (3 x y)}$

Step 8

Pull terms out from under the radical.

$18 x^{2} y^{4} (2 \sqrt{3 x y})$

Step 9

Multiply $2$ by $18$ .

$36 x^{2} y^{4} \sqrt{3 x y}$

Simplify square root of 54x^4y^3* square root of 72xy^6

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