Basic Math Examples

$\sqrt{\frac{2}{3}} + \sqrt{\frac{18}{17}}$

Step 1

Simplify each term.

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Step 1.1

Rewrite $\sqrt{\frac{2}{3}}$ as $\frac{\sqrt{2}}{\sqrt{3}}$ .

$\frac{\sqrt{2}}{\sqrt{3}} + \sqrt{\frac{18}{17}}$

Step 1.2

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

$\frac{\sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} + \sqrt{\frac{18}{17}}$

Step 1.3

Combine and simplify the denominator.

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Step 1.3.1

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

$\frac{\sqrt{2} \sqrt{3}}{\sqrt{3} \sqrt{3}} + \sqrt{\frac{18}{17}}$

Step 1.3.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} + \sqrt{\frac{18}{17}}$

Step 1.3.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} + \sqrt{\frac{18}{17}}$

Step 1.3.4

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

$\frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1 + 1}} + \sqrt{\frac{18}{17}}$

Step 1.3.5

Add $1$ and $1$ .

$\frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{2}} + \sqrt{\frac{18}{17}}$

Step 1.3.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\sqrt{2} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} + \sqrt{\frac{18}{17}}$

Step 1.3.6.2

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

$\frac{\sqrt{2} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} + \sqrt{\frac{18}{17}}$

Step 1.3.6.3

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

$\frac{\sqrt{2} \sqrt{3}}{3^{\frac{2}{2}}} + \sqrt{\frac{18}{17}}$

Step 1.3.6.4

Cancel the common factor of $2$ .

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Step 1.3.6.4.1

Cancel the common factor.

$\frac{\sqrt{2} \sqrt{3}}{3^{\frac{2}{2}}} + \sqrt{\frac{18}{17}}$

Step 1.3.6.4.2

Rewrite the expression.

$\frac{\sqrt{2} \sqrt{3}}{3^{1}} + \sqrt{\frac{18}{17}}$

Step 1.3.6.5

Evaluate the exponent.

$\frac{\sqrt{2} \sqrt{3}}{3} + \sqrt{\frac{18}{17}}$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Combine using the product rule for radicals.

$\frac{\sqrt{2 \cdot 3}}{3} + \sqrt{\frac{18}{17}}$

Step 1.4.2

Multiply $2$ by $3$ .

$\frac{\sqrt{6}}{3} + \sqrt{\frac{18}{17}}$

Step 1.5

Rewrite $\sqrt{\frac{18}{17}}$ as $\frac{\sqrt{18}}{\sqrt{17}}$ .

$\frac{\sqrt{6}}{3} + \frac{\sqrt{18}}{\sqrt{17}}$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 1.6.1.1

Factor $9$ out of $18$ .

$\frac{\sqrt{6}}{3} + \frac{\sqrt{9 (2)}}{\sqrt{17}}$

Step 1.6.1.2

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

$\frac{\sqrt{6}}{3} + \frac{\sqrt{3^{2} \cdot 2}}{\sqrt{17}}$

Step 1.6.2

Pull terms out from under the radical.

$\frac{\sqrt{6}}{3} + \frac{3 \sqrt{2}}{\sqrt{17}}$

Step 1.7

Multiply $\frac{3 \sqrt{2}}{\sqrt{17}}$ by $\frac{\sqrt{17}}{\sqrt{17}}$ .

$\frac{\sqrt{6}}{3} + \frac{3 \sqrt{2}}{\sqrt{17}} \cdot \frac{\sqrt{17}}{\sqrt{17}}$

Step 1.8

Combine and simplify the denominator.

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Step 1.8.1

Multiply $\frac{3 \sqrt{2}}{\sqrt{17}}$ by $\frac{\sqrt{17}}{\sqrt{17}}$ .

$\frac{\sqrt{6}}{3} + \frac{3 \sqrt{2} \sqrt{17}}{\sqrt{17} \sqrt{17}}$

Step 1.8.2

Raise $\sqrt{17}$ to the power of $1$ .

$\frac{\sqrt{6}}{3} + \frac{3 \sqrt{2} \sqrt{17}}{{\sqrt{17}}^{1} \sqrt{17}}$

Step 1.8.3

Raise $\sqrt{17}$ to the power of $1$ .

$\frac{\sqrt{6}}{3} + \frac{3 \sqrt{2} \sqrt{17}}{{\sqrt{17}}^{1} {\sqrt{17}}^{1}}$

Step 1.8.4

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

$\frac{\sqrt{6}}{3} + \frac{3 \sqrt{2} \sqrt{17}}{{\sqrt{17}}^{1 + 1}}$

Step 1.8.5

Add $1$ and $1$ .

$\frac{\sqrt{6}}{3} + \frac{3 \sqrt{2} \sqrt{17}}{{\sqrt{17}}^{2}}$

Step 1.8.6

Rewrite ${\sqrt{17}}^{2}$ as $17$ .

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Step 1.8.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{17}$ as $17^{\frac{1}{2}}$ .

$\frac{\sqrt{6}}{3} + \frac{3 \sqrt{2} \sqrt{17}}{{(17^{\frac{1}{2}})}^{2}}$

Step 1.8.6.2

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

$\frac{\sqrt{6}}{3} + \frac{3 \sqrt{2} \sqrt{17}}{17^{\frac{1}{2} \cdot 2}}$

Step 1.8.6.3

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

$\frac{\sqrt{6}}{3} + \frac{3 \sqrt{2} \sqrt{17}}{17^{\frac{2}{2}}}$

Step 1.8.6.4

Cancel the common factor of $2$ .

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Step 1.8.6.4.1

Cancel the common factor.

$\frac{\sqrt{6}}{3} + \frac{3 \sqrt{2} \sqrt{17}}{17^{\frac{2}{2}}}$

Step 1.8.6.4.2

Rewrite the expression.

$\frac{\sqrt{6}}{3} + \frac{3 \sqrt{2} \sqrt{17}}{17^{1}}$

Step 1.8.6.5

Evaluate the exponent.

$\frac{\sqrt{6}}{3} + \frac{3 \sqrt{2} \sqrt{17}}{17}$

Step 1.9

Simplify the numerator.

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Step 1.9.1

Combine using the product rule for radicals.

$\frac{\sqrt{6}}{3} + \frac{3 \sqrt{17 \cdot 2}}{17}$

Step 1.9.2

Multiply $17$ by $2$ .

$\frac{\sqrt{6}}{3} + \frac{3 \sqrt{34}}{17}$

Step 2

To write $\frac{\sqrt{6}}{3}$ as a fraction with a common denominator, multiply by $\frac{17}{17}$ .

$\frac{\sqrt{6}}{3} \cdot \frac{17}{17} + \frac{3 \sqrt{34}}{17}$

Step 3

To write $\frac{3 \sqrt{34}}{17}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\sqrt{6}}{3} \cdot \frac{17}{17} + \frac{3 \sqrt{34}}{17} \cdot \frac{3}{3}$

Step 4

Write each expression with a common denominator of $51$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{\sqrt{6}}{3}$ by $\frac{17}{17}$ .

$\frac{\sqrt{6} \cdot 17}{3 \cdot 17} + \frac{3 \sqrt{34}}{17} \cdot \frac{3}{3}$

Step 4.2

Multiply $3$ by $17$ .

$\frac{\sqrt{6} \cdot 17}{51} + \frac{3 \sqrt{34}}{17} \cdot \frac{3}{3}$

Step 4.3

Multiply $\frac{3 \sqrt{34}}{17}$ by $\frac{3}{3}$ .

$\frac{\sqrt{6} \cdot 17}{51} + \frac{3 \sqrt{34} \cdot 3}{17 \cdot 3}$

Step 4.4

Multiply $17$ by $3$ .

$\frac{\sqrt{6} \cdot 17}{51} + \frac{3 \sqrt{34} \cdot 3}{51}$

Step 5

Combine the numerators over the common denominator.

$\frac{\sqrt{6} \cdot 17 + 3 \sqrt{34} \cdot 3}{51}$

Step 6

Simplify the numerator.

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Step 6.1

Move $17$ to the left of $\sqrt{6}$ .

$\frac{17 \cdot \sqrt{6} + 3 \sqrt{34} \cdot 3}{51}$

Step 6.2

Multiply $3$ by $3$ .

$\frac{17 \sqrt{6} + 9 \sqrt{34}}{51}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{17 \sqrt{6} + 9 \sqrt{34}}{51}$

Decimal Form:

$1.84548809 \dots$