Evaluate ( square root of 32)/3+(4 square root of 2)/3-( square root of 2)/( square root of 9)

$\frac{\sqrt{32}}{3} + \frac{4 \sqrt{2}}{3} - \frac{\sqrt{2}}{\sqrt{9}}$

Step 1

Combine the numerators over the common denominator.

$\frac{\sqrt{32} + 4 \sqrt{2}}{3} + \frac{- \sqrt{2}}{\sqrt{9}}$

Step 2

Simplify each term.

Rewrite $32$ as $4^{2} \cdot 2$ .

Factor $16$ out of $32$ .

$\frac{\sqrt{16 (2)} + 4 \sqrt{2}}{3} + \frac{- \sqrt{2}}{\sqrt{9}}$

Rewrite $16$ as $4^{2}$ .

$\frac{\sqrt{4^{2} \cdot 2} + 4 \sqrt{2}}{3} + \frac{- \sqrt{2}}{\sqrt{9}}$

Pull terms out from under the radical.

$\frac{4 \sqrt{2} + 4 \sqrt{2}}{3} + \frac{- \sqrt{2}}{\sqrt{9}}$

Step 3

Add $4 \sqrt{2}$ and $4 \sqrt{2}$ .

$\frac{8 \sqrt{2}}{3} + \frac{- \sqrt{2}}{\sqrt{9}}$

Step 4

Simplify each term.

Simplify the denominator.

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

$\frac{8 \sqrt{2}}{3} + \frac{- \sqrt{2}}{\sqrt{3^{2}}}$

Pull terms out from under the radical, assuming positive real numbers.

$\frac{8 \sqrt{2}}{3} + \frac{- \sqrt{2}}{3}$

Move the negative in front of the fraction.

$\frac{8 \sqrt{2}}{3} - \frac{\sqrt{2}}{3}$

Step 5

Simplify terms.

Combine the numerators over the common denominator.

$\frac{8 \sqrt{2} - \sqrt{2}}{3}$

Subtract $\sqrt{2}$ from $8 \sqrt{2}$ .

$\frac{7 \sqrt{2}}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{7 \sqrt{2}}{3}$

Decimal Form:

$3.29983164 \dots$