Simplify the Radical Expression y square root of 28x^2y+2 square root of 18xy^3-4 square root of 27x^2y^2

$y \sqrt{28 x^{2} y} + 2 \sqrt{18 x y^{3}} - 4 \sqrt{27 x^{2} y^{2}}$

Step 1

Simplify each term.

Rewrite $28 x^{2} y$ as ${(2 x)}^{2} \cdot (7 y)$ .

Factor $4$ out of $28$ .

$y \sqrt{4 (7) x^{2} y} + 2 \sqrt{18 x y^{3}} - 4 \sqrt{27 x^{2} y^{2}}$

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

$y \sqrt{2^{2} \cdot 7 x^{2} y} + 2 \sqrt{18 x y^{3}} - 4 \sqrt{27 x^{2} y^{2}}$

Move $7$ .

$y \sqrt{2^{2} x^{2} \cdot 7 y} + 2 \sqrt{18 x y^{3}} - 4 \sqrt{27 x^{2} y^{2}}$

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

$y \sqrt{{(2 x)}^{2} \cdot 7 y} + 2 \sqrt{18 x y^{3}} - 4 \sqrt{27 x^{2} y^{2}}$

Add parentheses.

$y \sqrt{{(2 x)}^{2} \cdot (7 y)} + 2 \sqrt{18 x y^{3}} - 4 \sqrt{27 x^{2} y^{2}}$

Pull terms out from under the radical.

$y (| 2 x | \sqrt{7 y}) + 2 \sqrt{18 x y^{3}} - 4 \sqrt{27 x^{2} y^{2}}$

Remove non-negative terms from the absolute value.

$y (2 | x | \sqrt{7 y}) + 2 \sqrt{18 x y^{3}} - 4 \sqrt{27 x^{2} y^{2}}$

Rewrite using the commutative property of multiplication.

$2 y (| x | \sqrt{7 y}) + 2 \sqrt{18 x y^{3}} - 4 \sqrt{27 x^{2} y^{2}}$

Rewrite $18 x y^{3}$ as ${(3 y)}^{2} \cdot (2 x y)$ .

Factor $9$ out of $18$ .

$2 y (| x | \sqrt{7 y}) + 2 \sqrt{9 (2) x y^{3}} - 4 \sqrt{27 x^{2} y^{2}}$

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

$2 y (| x | \sqrt{7 y}) + 2 \sqrt{3^{2} \cdot 2 x y^{3}} - 4 \sqrt{27 x^{2} y^{2}}$

Factor out $y^{2}$ .

$2 y (| x | \sqrt{7 y}) + 2 \sqrt{3^{2} \cdot 2 x (y^{2} y)} - 4 \sqrt{27 x^{2} y^{2}}$

Move $x$ .

$2 y (| x | \sqrt{7 y}) + 2 \sqrt{3^{2} \cdot 2 y^{2} x y} - 4 \sqrt{27 x^{2} y^{2}}$

Move $2$ .

$2 y (| x | \sqrt{7 y}) + 2 \sqrt{3^{2} y^{2} \cdot 2 x y} - 4 \sqrt{27 x^{2} y^{2}}$

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

$2 y (| x | \sqrt{7 y}) + 2 \sqrt{{(3 y)}^{2} \cdot 2 x y} - 4 \sqrt{27 x^{2} y^{2}}$

Add parentheses.

$2 y (| x | \sqrt{7 y}) + 2 \sqrt{{(3 y)}^{2} \cdot 2 (x y)} - 4 \sqrt{27 x^{2} y^{2}}$

Add parentheses.

$2 y (| x | \sqrt{7 y}) + 2 \sqrt{{(3 y)}^{2} \cdot (2 x y)} - 4 \sqrt{27 x^{2} y^{2}}$

Pull terms out from under the radical.

$2 y (| x | \sqrt{7 y}) + 2 (| 3 y | \sqrt{2 x y}) - 4 \sqrt{27 x^{2} y^{2}}$

Remove non-negative terms from the absolute value.

$2 y (| x | \sqrt{7 y}) + 2 (3 | y | \sqrt{2 x y}) - 4 \sqrt{27 x^{2} y^{2}}$

Multiply $3$ by $2$ .

$2 y (| x | \sqrt{7 y}) + 6 (| y | \sqrt{2 x y}) - 4 \sqrt{27 x^{2} y^{2}}$

Rewrite $27 x^{2} y^{2}$ as ${(3 x y)}^{2} \cdot 3$ .

Factor $9$ out of $27$ .

$2 y (| x | \sqrt{7 y}) + 6 | y | \sqrt{2 x y} - 4 \sqrt{9 (3) x^{2} y^{2}}$

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

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

Move $3$ .

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

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

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

Pull terms out from under the radical.

$2 y (| x | \sqrt{7 y}) + 6 | y | \sqrt{2 x y} - 4 (| 3 x y | \sqrt{3})$

Remove non-negative terms from the absolute value.

$2 y (| x | \sqrt{7 y}) + 6 | y | \sqrt{2 x y} - 4 (3 | x y | \sqrt{3})$

Multiply $3$ by $- 4$ .

$2 y (| x | \sqrt{7 y}) + 6 | y | \sqrt{2 x y} - 12 | x y | \sqrt{3}$

Step 2

Reorder factors in $2 y (| x | \sqrt{7 y}) + 6 | y | \sqrt{2 x y} - 12 | x y | \sqrt{3}$ .

$2 y | x | \sqrt{7 y} + 6 | y | \sqrt{2 x y} - 12 | x y | \sqrt{3}$

Simplify the Radical Expression y square root of 28x^2y+2 square root of 18xy^3-4 square root of 27x^2y^2

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