Simplify 1/3* square root of 45-1/2* square root of 12+ square root of 20+2/3* square root of 27

$\frac{1}{3} \cdot \sqrt{45} - \frac{1}{2} \cdot \sqrt{12} + \sqrt{20} + \frac{2}{3} \cdot \sqrt{27}$

Step 1

Simplify each term.

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

Factor $9$ out of $45$ .

$\frac{1}{3} \cdot \sqrt{9 (5)} - \frac{1}{2} \cdot \sqrt{12} + \sqrt{20} + \frac{2}{3} \cdot \sqrt{27}$

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

$\frac{1}{3} \cdot \sqrt{3^{2} \cdot 5} - \frac{1}{2} \cdot \sqrt{12} + \sqrt{20} + \frac{2}{3} \cdot \sqrt{27}$

Pull terms out from under the radical.

$\frac{1}{3} \cdot (3 \sqrt{5}) - \frac{1}{2} \cdot \sqrt{12} + \sqrt{20} + \frac{2}{3} \cdot \sqrt{27}$

Cancel the common factor of $3$ .

Factor $3$ out of $3 \sqrt{5}$ .

$\frac{1}{3} \cdot (3 (\sqrt{5})) - \frac{1}{2} \cdot \sqrt{12} + \sqrt{20} + \frac{2}{3} \cdot \sqrt{27}$

Cancel the common factor.

$\frac{1}{3} \cdot (3 \sqrt{5}) - \frac{1}{2} \cdot \sqrt{12} + \sqrt{20} + \frac{2}{3} \cdot \sqrt{27}$

Rewrite the expression.

$\sqrt{5} - \frac{1}{2} \cdot \sqrt{12} + \sqrt{20} + \frac{2}{3} \cdot \sqrt{27}$

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

Factor $4$ out of $12$ .

$\sqrt{5} - \frac{1}{2} \cdot \sqrt{4 (3)} + \sqrt{20} + \frac{2}{3} \cdot \sqrt{27}$

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

$\sqrt{5} - \frac{1}{2} \cdot \sqrt{2^{2} \cdot 3} + \sqrt{20} + \frac{2}{3} \cdot \sqrt{27}$

Pull terms out from under the radical.

$\sqrt{5} - \frac{1}{2} \cdot (2 \sqrt{3}) + \sqrt{20} + \frac{2}{3} \cdot \sqrt{27}$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$\sqrt{5} + \frac{- 1}{2} \cdot (2 \sqrt{3}) + \sqrt{20} + \frac{2}{3} \cdot \sqrt{27}$

Factor $2$ out of $2 \sqrt{3}$ .

$\sqrt{5} + \frac{- 1}{2} \cdot (2 (\sqrt{3})) + \sqrt{20} + \frac{2}{3} \cdot \sqrt{27}$

Cancel the common factor.

$\sqrt{5} + \frac{- 1}{2} \cdot (2 \sqrt{3}) + \sqrt{20} + \frac{2}{3} \cdot \sqrt{27}$

Rewrite the expression.

$\sqrt{5} - 1 \cdot \sqrt{3} + \sqrt{20} + \frac{2}{3} \cdot \sqrt{27}$

Rewrite $- 1 \sqrt{3}$ as $- \sqrt{3}$ .

$\sqrt{5} - \sqrt{3} + \sqrt{20} + \frac{2}{3} \cdot \sqrt{27}$

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

Factor $4$ out of $20$ .

$\sqrt{5} - \sqrt{3} + \sqrt{4 (5)} + \frac{2}{3} \cdot \sqrt{27}$

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

$\sqrt{5} - \sqrt{3} + \sqrt{2^{2} \cdot 5} + \frac{2}{3} \cdot \sqrt{27}$

Pull terms out from under the radical.

$\sqrt{5} - \sqrt{3} + 2 \sqrt{5} + \frac{2}{3} \cdot \sqrt{27}$

Rewrite $27$ as $3^{2} \cdot 3$ .

Factor $9$ out of $27$ .

$\sqrt{5} - \sqrt{3} + 2 \sqrt{5} + \frac{2}{3} \cdot \sqrt{9 (3)}$

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

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

Pull terms out from under the radical.

$\sqrt{5} - \sqrt{3} + 2 \sqrt{5} + \frac{2}{3} \cdot (3 \sqrt{3})$

Cancel the common factor of $3$ .

Factor $3$ out of $3 \sqrt{3}$ .

$\sqrt{5} - \sqrt{3} + 2 \sqrt{5} + \frac{2}{3} \cdot (3 (\sqrt{3}))$

Cancel the common factor.

$\sqrt{5} - \sqrt{3} + 2 \sqrt{5} + \frac{2}{3} \cdot (3 \sqrt{3})$

Rewrite the expression.

$\sqrt{5} - \sqrt{3} + 2 \sqrt{5} + 2 \cdot \sqrt{3}$

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

Step 2

Simplify by adding terms.

Add $\sqrt{5}$ and $2 \sqrt{5}$ .

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

Add $- \sqrt{3}$ and $2 \sqrt{3}$ .

$3 \sqrt{5} + \sqrt{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$3 \sqrt{5} + \sqrt{3}$

Decimal Form:

$8.44025474 \dots$

Simplify 1/3* square root of 45-1/2* square root of 12+ square root of 20+2/3* square root of 27

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