Algebra Examples

$\frac{\sqrt{50}}{3} + \sqrt{108} + \frac{1}{\sqrt{2}}$

Step 1

Simplify each term.

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Step 1.1

Simplify the numerator.

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Step 1.1.1

Rewrite $50$ as $5^{2} \cdot 2$ .

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Step 1.1.1.1

Factor $25$ out of $50$ .

$\frac{\sqrt{25 (2)}}{3} + \sqrt{108} + \frac{1}{\sqrt{2}}$

Step 1.1.1.2

Rewrite $25$ as $5^{2}$ .

$\frac{\sqrt{5^{2} \cdot 2}}{3} + \sqrt{108} + \frac{1}{\sqrt{2}}$

Step 1.1.2

Pull terms out from under the radical.

$\frac{5 \sqrt{2}}{3} + \sqrt{108} + \frac{1}{\sqrt{2}}$

Step 1.2

Rewrite $108$ as $6^{2} \cdot 3$ .

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Step 1.2.1

Factor $36$ out of $108$ .

$\frac{5 \sqrt{2}}{3} + \sqrt{36 (3)} + \frac{1}{\sqrt{2}}$

Step 1.2.2

Rewrite $36$ as $6^{2}$ .

$\frac{5 \sqrt{2}}{3} + \sqrt{6^{2} \cdot 3} + \frac{1}{\sqrt{2}}$

Step 1.3

Pull terms out from under the radical.

$\frac{5 \sqrt{2}}{3} + 6 \sqrt{3} + \frac{1}{\sqrt{2}}$

Step 1.4

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

$\frac{5 \sqrt{2}}{3} + 6 \sqrt{3} + \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 1.5

Combine and simplify the denominator.

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Step 1.5.1

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

$\frac{5 \sqrt{2}}{3} + 6 \sqrt{3} + \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 1.5.2

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

$\frac{5 \sqrt{2}}{3} + 6 \sqrt{3} + \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 1.5.3

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

$\frac{5 \sqrt{2}}{3} + 6 \sqrt{3} + \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 1.5.4

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

$\frac{5 \sqrt{2}}{3} + 6 \sqrt{3} + \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 1.5.5

Add $1$ and $1$ .

$\frac{5 \sqrt{2}}{3} + 6 \sqrt{3} + \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 1.5.6

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

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Step 1.5.6.1

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

$\frac{5 \sqrt{2}}{3} + 6 \sqrt{3} + \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 1.5.6.2

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

$\frac{5 \sqrt{2}}{3} + 6 \sqrt{3} + \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 1.5.6.3

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

$\frac{5 \sqrt{2}}{3} + 6 \sqrt{3} + \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.5.6.4

Cancel the common factor of $2$ .

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Step 1.5.6.4.1

Cancel the common factor.

$\frac{5 \sqrt{2}}{3} + 6 \sqrt{3} + \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.5.6.4.2

Rewrite the expression.

$\frac{5 \sqrt{2}}{3} + 6 \sqrt{3} + \frac{\sqrt{2}}{2^{1}}$

Step 1.5.6.5

Evaluate the exponent.

$\frac{5 \sqrt{2}}{3} + 6 \sqrt{3} + \frac{\sqrt{2}}{2}$

Step 2

Find the common denominator.

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Step 2.1

Multiply $\frac{5 \sqrt{2}}{3}$ by $\frac{2}{2}$ .

$\frac{5 \sqrt{2}}{3} \cdot \frac{2}{2} + 6 \sqrt{3} + \frac{\sqrt{2}}{2}$

Step 2.2

Multiply $\frac{5 \sqrt{2}}{3}$ by $\frac{2}{2}$ .

$\frac{5 \sqrt{2} \cdot 2}{3 \cdot 2} + 6 \sqrt{3} + \frac{\sqrt{2}}{2}$

Step 2.3

Write $6 \sqrt{3}$ as a fraction with denominator $1$ .

$\frac{5 \sqrt{2} \cdot 2}{3 \cdot 2} + \frac{6 \sqrt{3}}{1} + \frac{\sqrt{2}}{2}$

Step 2.4

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

$\frac{5 \sqrt{2} \cdot 2}{3 \cdot 2} + \frac{6 \sqrt{3}}{1} \cdot \frac{6}{6} + \frac{\sqrt{2}}{2}$

Step 2.5

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

$\frac{5 \sqrt{2} \cdot 2}{3 \cdot 2} + \frac{6 \sqrt{3} \cdot 6}{6} + \frac{\sqrt{2}}{2}$

Step 2.6

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

$\frac{5 \sqrt{2} \cdot 2}{3 \cdot 2} + \frac{6 \sqrt{3} \cdot 6}{6} + \frac{\sqrt{2}}{2} \cdot \frac{3}{3}$

Step 2.7

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

$\frac{5 \sqrt{2} \cdot 2}{3 \cdot 2} + \frac{6 \sqrt{3} \cdot 6}{6} + \frac{\sqrt{2} \cdot 3}{2 \cdot 3}$

Step 2.8

Reorder the factors of $3 \cdot 2$ .

$\frac{5 \sqrt{2} \cdot 2}{2 \cdot 3} + \frac{6 \sqrt{3} \cdot 6}{6} + \frac{\sqrt{2} \cdot 3}{2 \cdot 3}$

Step 2.9

Multiply $2$ by $3$ .

$\frac{5 \sqrt{2} \cdot 2}{6} + \frac{6 \sqrt{3} \cdot 6}{6} + \frac{\sqrt{2} \cdot 3}{2 \cdot 3}$

Step 2.10

Multiply $2$ by $3$ .

$\frac{5 \sqrt{2} \cdot 2}{6} + \frac{6 \sqrt{3} \cdot 6}{6} + \frac{\sqrt{2} \cdot 3}{6}$

Step 3

Combine the numerators over the common denominator.

$\frac{5 \sqrt{2} \cdot 2 + 6 \sqrt{3} \cdot 6 + \sqrt{2} \cdot 3}{6}$

Step 4

Simplify each term.

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Step 4.1

Multiply $2$ by $5$ .

$\frac{10 \sqrt{2} + 6 \sqrt{3} \cdot 6 + \sqrt{2} \cdot 3}{6}$

Step 4.2

Multiply $6$ by $6$ .

$\frac{10 \sqrt{2} + 36 \sqrt{3} + \sqrt{2} \cdot 3}{6}$

Step 4.3

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

$\frac{10 \sqrt{2} + 36 \sqrt{3} + 3 \sqrt{2}}{6}$

Step 5

Add $10 \sqrt{2}$ and $3 \sqrt{2}$ .

$\frac{13 \sqrt{2} + 36 \sqrt{3}}{6}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{13 \sqrt{2} + 36 \sqrt{3}}{6}$

Decimal Form:

$13.45643423 \dots$