Simplify 3 square root of 90+4 square root of 20+ square root of 162

$3 \sqrt{90} + 4 \sqrt{20} + \sqrt{162}$

Step 1

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

Factor $9$ out of $90$ .

$3 \sqrt{9 (10)} + 4 \sqrt{20} + \sqrt{162}$

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

$3 \sqrt{3^{2} \cdot 10} + 4 \sqrt{20} + \sqrt{162}$

Step 2

Pull terms out from under the radical.

$3 (3 \sqrt{10}) + 4 \sqrt{20} + \sqrt{162}$

Step 3

Multiply $3$ by $3$ .

$9 \sqrt{10} + 4 \sqrt{20} + \sqrt{162}$

Step 4

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

Factor $4$ out of $20$ .

$9 \sqrt{10} + 4 \sqrt{4 (5)} + \sqrt{162}$

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

$9 \sqrt{10} + 4 \sqrt{2^{2} \cdot 5} + \sqrt{162}$

Step 5

Pull terms out from under the radical.

$9 \sqrt{10} + 4 (2 \sqrt{5}) + \sqrt{162}$

Step 6

Multiply $2$ by $4$ .

$9 \sqrt{10} + 8 \sqrt{5} + \sqrt{162}$

Step 7

Rewrite $162$ as $9^{2} \cdot 2$ .

Factor $81$ out of $162$ .

$9 \sqrt{10} + 8 \sqrt{5} + \sqrt{81 (2)}$

Rewrite $81$ as $9^{2}$ .

$9 \sqrt{10} + 8 \sqrt{5} + \sqrt{9^{2} \cdot 2}$

Step 8

Pull terms out from under the radical.

$9 \sqrt{10} + 8 \sqrt{5} + 9 \sqrt{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$9 \sqrt{10} + 8 \sqrt{5} + 9 \sqrt{2}$

Decimal Form:

$59.07696482 \dots$