Precalculus Examples

$\sqrt{R + \frac{\sqrt{2}}{5}}$

Step 1

To write $R$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\sqrt{R \cdot \frac{5}{5} + \frac{\sqrt{2}}{5}}$

Step 2

Simplify terms.

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Step 2.1

Combine $R$ and $\frac{5}{5}$ .

$\sqrt{\frac{R \cdot 5}{5} + \frac{\sqrt{2}}{5}}$

Step 2.2

Combine the numerators over the common denominator.

$\sqrt{\frac{R \cdot 5 + \sqrt{2}}{5}}$

Step 3

Move $5$ to the left of $R$ .

$\sqrt{\frac{5 R + \sqrt{2}}{5}}$

Step 4

Rewrite $\sqrt{\frac{5 R + \sqrt{2}}{5}}$ as $\frac{\sqrt{5 R + \sqrt{2}}}{\sqrt{5}}$ .

$\frac{\sqrt{5 R + \sqrt{2}}}{\sqrt{5}}$

Step 5

Multiply $\frac{\sqrt{5 R + \sqrt{2}}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\sqrt{5 R + \sqrt{2}}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{\sqrt{5 R + \sqrt{2}}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\sqrt{5 R + \sqrt{2}} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 6.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{\sqrt{5 R + \sqrt{2}} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 6.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{\sqrt{5 R + \sqrt{2}} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 6.4

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

$\frac{\sqrt{5 R + \sqrt{2}} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 6.5

Add $1$ and $1$ .

$\frac{\sqrt{5 R + \sqrt{2}} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 6.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

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

Step 6.6.2

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

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

Step 6.6.3

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

$\frac{\sqrt{5 R + \sqrt{2}} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 6.6.4

Cancel the common factor of $2$ .

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Step 6.6.4.1

Cancel the common factor.

$\frac{\sqrt{5 R + \sqrt{2}} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 6.6.4.2

Rewrite the expression.

$\frac{\sqrt{5 R + \sqrt{2}} \sqrt{5}}{5^{1}}$

Step 6.6.5

Evaluate the exponent.

$\frac{\sqrt{5 R + \sqrt{2}} \sqrt{5}}{5}$

Step 7

Combine using the product rule for radicals.

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