Simplify the Radical Expression (3 square root of 5)( square root of 6)( square root of 12)

$(3 \sqrt{5}) (\sqrt{6}) (\sqrt{12})$

Step 1

Multiply $(3 \sqrt{5}) (\sqrt{6})$ .

Combine using the product rule for radicals.

$3 \sqrt{6 \cdot 5} \sqrt{12}$

Multiply $6$ by $5$ .

$3 \sqrt{30} \sqrt{12}$

Step 2

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

Factor $4$ out of $12$ .

$3 \sqrt{30} \sqrt{4 (3)}$

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

$3 \sqrt{30} \sqrt{2^{2} \cdot 3}$

Step 3

Pull terms out from under the radical.

$3 \sqrt{30} (| 2 | \sqrt{3})$

Step 4

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

$3 \sqrt{30} (2 \sqrt{3})$

Step 5

Multiply $3 \sqrt{30} (2 \sqrt{3})$ .

Multiply $2$ by $3$ .

$6 \sqrt{30} \sqrt{3}$

Combine using the product rule for radicals.

$6 \sqrt{3 \cdot 30}$

Multiply $3$ by $30$ .

$6 \sqrt{90}$

Step 6

Rewrite $90$ as $3^{2} \cdot 10$ .

Factor $9$ out of $90$ .

$6 \sqrt{9 (10)}$

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

$6 \sqrt{3^{2} \cdot 10}$

Step 7

Pull terms out from under the radical.

$6 (| 3 | \sqrt{10})$

Step 8

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

$6 (3 \sqrt{10})$

Step 9

Multiply $3$ by $6$ .

$18 \sqrt{10}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$18 \sqrt{10}$

Decimal Form:

$56.92099788 \dots$

Simplify the Radical Expression (3 square root of 5)( square root of 6)( square root of 12)

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