Simplify the Radical Expression 3 square root of 20-2 square root of 24+ square root of 180- square root of 54

$3 \sqrt{20} - 2 \sqrt{24} + \sqrt{180} - \sqrt{54}$

Step 1

Simplify each term.

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

Factor $4$ out of $20$ .

$3 \sqrt{4 (5)} - 2 \sqrt{24} + \sqrt{180} - \sqrt{54}$

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

$3 \sqrt{2^{2} \cdot 5} - 2 \sqrt{24} + \sqrt{180} - \sqrt{54}$

Pull terms out from under the radical.

$3 (| 2 | \sqrt{5}) - 2 \sqrt{24} + \sqrt{180} - \sqrt{54}$

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

$3 (2 \sqrt{5}) - 2 \sqrt{24} + \sqrt{180} - \sqrt{54}$

Multiply $2$ by $3$ .

$6 \sqrt{5} - 2 \sqrt{24} + \sqrt{180} - \sqrt{54}$

Rewrite $24$ as $2^{2} \cdot 6$ .

Factor $4$ out of $24$ .

$6 \sqrt{5} - 2 \sqrt{4 (6)} + \sqrt{180} - \sqrt{54}$

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

$6 \sqrt{5} - 2 \sqrt{2^{2} \cdot 6} + \sqrt{180} - \sqrt{54}$

Pull terms out from under the radical.

$6 \sqrt{5} - 2 (| 2 | \sqrt{6}) + \sqrt{180} - \sqrt{54}$

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

$6 \sqrt{5} - 2 (2 \sqrt{6}) + \sqrt{180} - \sqrt{54}$

Multiply $2$ by $- 2$ .

$6 \sqrt{5} - 4 \sqrt{6} + \sqrt{180} - \sqrt{54}$

Rewrite $180$ as $6^{2} \cdot 5$ .

Factor $36$ out of $180$ .

$6 \sqrt{5} - 4 \sqrt{6} + \sqrt{36 (5)} - \sqrt{54}$

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

$6 \sqrt{5} - 4 \sqrt{6} + \sqrt{6^{2} \cdot 5} - \sqrt{54}$

Pull terms out from under the radical.

$6 \sqrt{5} - 4 \sqrt{6} + | 6 | \sqrt{5} - \sqrt{54}$

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

$6 \sqrt{5} - 4 \sqrt{6} + 6 \sqrt{5} - \sqrt{54}$

Rewrite $54$ as $3^{2} \cdot 6$ .

Factor $9$ out of $54$ .

$6 \sqrt{5} - 4 \sqrt{6} + 6 \sqrt{5} - \sqrt{9 (6)}$

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

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

Pull terms out from under the radical.

$6 \sqrt{5} - 4 \sqrt{6} + 6 \sqrt{5} - (| 3 | \sqrt{6})$

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$6 \sqrt{5} - 4 \sqrt{6} + 6 \sqrt{5} - (3 \sqrt{6})$

Multiply $3$ by $- 1$ .

$6 \sqrt{5} - 4 \sqrt{6} + 6 \sqrt{5} - 3 \sqrt{6}$

Step 2

Add $6 \sqrt{5}$ and $6 \sqrt{5}$ .

$12 \sqrt{5} - 4 \sqrt{6} - 3 \sqrt{6}$

Step 3

Subtract $3 \sqrt{6}$ from $- 4 \sqrt{6}$ .

$12 \sqrt{5} - 7 \sqrt{6}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$12 \sqrt{5} - 7 \sqrt{6}$

Decimal Form:

$9.68638753 \dots$

Simplify the Radical Expression 3 square root of 20-2 square root of 24+ square root of 180- square root of 54

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