Simplify ( square root of 128)/3+ square root of 243+1/( square root of 2)

$\frac{\sqrt{128}}{3} + \sqrt{243} + \frac{1}{\sqrt{2}}$

Simplify each term.

Simplify the numerator.

Rewrite $128$ as $8^{2} \cdot 2$ .

Factor $64$ out of $128$ .

$\frac{\sqrt{64 (2)}}{3} + \sqrt{243} + \frac{1}{\sqrt{2}}$

Rewrite $64$ as $8^{2}$ .

$\frac{\sqrt{8^{2} \cdot 2}}{3} + \sqrt{243} + \frac{1}{\sqrt{2}}$

Pull terms out from under the radical.

$\frac{8 \sqrt{2}}{3} + \sqrt{243} + \frac{1}{\sqrt{2}}$

Rewrite $243$ as $9^{2} \cdot 3$ .

Factor $81$ out of $243$ .

$\frac{8 \sqrt{2}}{3} + \sqrt{81 (3)} + \frac{1}{\sqrt{2}}$

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

$\frac{8 \sqrt{2}}{3} + \sqrt{9^{2} \cdot 3} + \frac{1}{\sqrt{2}}$

Pull terms out from under the radical.

$\frac{8 \sqrt{2}}{3} + 9 \sqrt{3} + \frac{1}{\sqrt{2}}$

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{8 \sqrt{2}}{3} + 9 \sqrt{3} + \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{8 \sqrt{2}}{3} + 9 \sqrt{3} + \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{8 \sqrt{2}}{3} + 9 \sqrt{3} + \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{8 \sqrt{2}}{3} + 9 \sqrt{3} + \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

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

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{8 \sqrt{2}}{3} + 9 \sqrt{3} + \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{8 \sqrt{2}}{3} + 9 \sqrt{3} + \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{8 \sqrt{2}}{3} + 9 \sqrt{3} + \frac{\sqrt{2}}{2^{1}}$

Evaluate the exponent.

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

Find the common denominator.

Multiply $\frac{8 \sqrt{2}}{3}$ by $\frac{2}{2}$ .

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

Multiply $\frac{8 \sqrt{2}}{3}$ by $\frac{2}{2}$ .

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

Write $9 \sqrt{3}$ as a fraction with denominator $1$ .

$\frac{8 \sqrt{2} \cdot 2}{3 \cdot 2} + \frac{9 \sqrt{3}}{1} + \frac{\sqrt{2}}{2}$

Multiply $\frac{9 \sqrt{3}}{1}$ by $\frac{6}{6}$ .

$\frac{8 \sqrt{2} \cdot 2}{3 \cdot 2} + \frac{9 \sqrt{3}}{1} \cdot \frac{6}{6} + \frac{\sqrt{2}}{2}$

Multiply $\frac{9 \sqrt{3}}{1}$ by $\frac{6}{6}$ .

$\frac{8 \sqrt{2} \cdot 2}{3 \cdot 2} + \frac{9 \sqrt{3} \cdot 6}{6} + \frac{\sqrt{2}}{2}$

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{3}{3}$ .

$\frac{8 \sqrt{2} \cdot 2}{3 \cdot 2} + \frac{9 \sqrt{3} \cdot 6}{6} + \frac{\sqrt{2}}{2} \cdot \frac{3}{3}$

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{3}{3}$ .

$\frac{8 \sqrt{2} \cdot 2}{3 \cdot 2} + \frac{9 \sqrt{3} \cdot 6}{6} + \frac{\sqrt{2} \cdot 3}{2 \cdot 3}$

Reorder the factors of $3 \cdot 2$ .

$\frac{8 \sqrt{2} \cdot 2}{2 \cdot 3} + \frac{9 \sqrt{3} \cdot 6}{6} + \frac{\sqrt{2} \cdot 3}{2 \cdot 3}$

Multiply $2$ by $3$ .

$\frac{8 \sqrt{2} \cdot 2}{6} + \frac{9 \sqrt{3} \cdot 6}{6} + \frac{\sqrt{2} \cdot 3}{2 \cdot 3}$

Multiply $2$ by $3$ .

$\frac{8 \sqrt{2} \cdot 2}{6} + \frac{9 \sqrt{3} \cdot 6}{6} + \frac{\sqrt{2} \cdot 3}{6}$

Combine the numerators over the common denominator.

$\frac{8 \sqrt{2} \cdot 2 + 9 \sqrt{3} \cdot 6 + \sqrt{2} \cdot 3}{6}$

Simplify each term.

Multiply $2$ by $8$ .

$\frac{16 \sqrt{2} + 9 \sqrt{3} \cdot 6 + \sqrt{2} \cdot 3}{6}$

Multiply $6$ by $9$ .

$\frac{16 \sqrt{2} + 54 \sqrt{3} + \sqrt{2} \cdot 3}{6}$

Move $3$ to the left of $\sqrt{2}$ .

$\frac{16 \sqrt{2} + 54 \sqrt{3} + 3 \sqrt{2}}{6}$

Add $16 \sqrt{2}$ and $3 \sqrt{2}$ .

$\frac{19 \sqrt{2} + 54 \sqrt{3}}{6}$

The result can be shown in multiple forms.

Exact Form:

$\frac{19 \sqrt{2} + 54 \sqrt{3}}{6}$

Decimal Form:

$20.06680021 \dots$

Simplify ( square root of 128)/3+ square root of 243+1/( square root of 2)

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