Simplify the Radical Expression 5 square root of 20-2 square root of 75+4 square root of 12

$5 \sqrt{20} - 2 \sqrt{75} + 4 \sqrt{12}$

Step 1

Simplify each term.

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

Factor $4$ out of $20$ .

$5 \sqrt{4 (5)} - 2 \sqrt{75} + 4 \sqrt{12}$

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

$5 \sqrt{2^{2} \cdot 5} - 2 \sqrt{75} + 4 \sqrt{12}$

Pull terms out from under the radical.

$5 (| 2 | \sqrt{5}) - 2 \sqrt{75} + 4 \sqrt{12}$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$5 (2 \sqrt{5}) - 2 \sqrt{75} + 4 \sqrt{12}$

Multiply $2$ by $5$ .

$10 \sqrt{5} - 2 \sqrt{75} + 4 \sqrt{12}$

Rewrite $75$ as $5^{2} \cdot 3$ .

Factor $25$ out of $75$ .

$10 \sqrt{5} - 2 \sqrt{25 (3)} + 4 \sqrt{12}$

Rewrite $25$ as $5^{2}$ .

$10 \sqrt{5} - 2 \sqrt{5^{2} \cdot 3} + 4 \sqrt{12}$

Pull terms out from under the radical.

$10 \sqrt{5} - 2 (| 5 | \sqrt{3}) + 4 \sqrt{12}$

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

$10 \sqrt{5} - 2 (5 \sqrt{3}) + 4 \sqrt{12}$

Multiply $5$ by $- 2$ .

$10 \sqrt{5} - 10 \sqrt{3} + 4 \sqrt{12}$

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$10 \sqrt{5} - 10 \sqrt{3} + 4 \sqrt{4 (3)}$

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

$10 \sqrt{5} - 10 \sqrt{3} + 4 \sqrt{2^{2} \cdot 3}$

Pull terms out from under the radical.

$10 \sqrt{5} - 10 \sqrt{3} + 4 (| 2 | \sqrt{3})$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$10 \sqrt{5} - 10 \sqrt{3} + 4 (2 \sqrt{3})$

Multiply $2$ by $4$ .

$10 \sqrt{5} - 10 \sqrt{3} + 8 \sqrt{3}$

Step 2

Add $- 10 \sqrt{3}$ and $8 \sqrt{3}$ .

$10 \sqrt{5} - 2 \sqrt{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$10 \sqrt{5} - 2 \sqrt{3}$

Decimal Form:

$18.89657815 \dots$