Simplify the Radical Expression 6 square root of 45y^2-4 square root of 20y^2

$6 \sqrt{45 y^{2}} - 4 \sqrt{20 y^{2}}$

Step 1

Simplify each term.

Rewrite $45 y^{2}$ as ${(3 y)}^{2} \cdot 5$ .

Factor $9$ out of $45$ .

$6 \sqrt{9 (5) y^{2}} - 4 \sqrt{20 y^{2}}$

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

$6 \sqrt{3^{2} \cdot 5 y^{2}} - 4 \sqrt{20 y^{2}}$

Move $5$ .

$6 \sqrt{3^{2} y^{2} \cdot 5} - 4 \sqrt{20 y^{2}}$

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

$6 \sqrt{{(3 y)}^{2} \cdot 5} - 4 \sqrt{20 y^{2}}$

Pull terms out from under the radical.

$6 (| 3 y | \sqrt{5}) - 4 \sqrt{20 y^{2}}$

Remove non-negative terms from the absolute value.

$6 (3 | y | \sqrt{5}) - 4 \sqrt{20 y^{2}}$

Multiply $3$ by $6$ .

$18 (| y | \sqrt{5}) - 4 \sqrt{20 y^{2}}$

Rewrite $20 y^{2}$ as ${(2 y)}^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$18 | y | \sqrt{5} - 4 \sqrt{4 (5) y^{2}}$

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

$18 | y | \sqrt{5} - 4 \sqrt{2^{2} \cdot 5 y^{2}}$

Move $5$ .

$18 | y | \sqrt{5} - 4 \sqrt{2^{2} y^{2} \cdot 5}$

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

$18 | y | \sqrt{5} - 4 \sqrt{{(2 y)}^{2} \cdot 5}$

Pull terms out from under the radical.

$18 | y | \sqrt{5} - 4 (| 2 y | \sqrt{5})$

Remove non-negative terms from the absolute value.

$18 | y | \sqrt{5} - 4 (2 | y | \sqrt{5})$

Multiply $2$ by $- 4$ .

$18 | y | \sqrt{5} - 8 | y | \sqrt{5}$

Step 2

Subtract $8 | y | \sqrt{5}$ from $18 | y | \sqrt{5}$ .

$10 | y | \sqrt{5}$