Basic Math Examples

$\frac{\sqrt{r}}{3 - \sqrt{r}}$

Step 1

Multiply $\frac{\sqrt{r}}{3 - \sqrt{r}}$ by $\frac{3 + \sqrt{r}}{3 + \sqrt{r}}$ .

$\frac{\sqrt{r}}{3 - \sqrt{r}} \cdot \frac{3 + \sqrt{r}}{3 + \sqrt{r}}$

Step 2

Multiply $\frac{\sqrt{r}}{3 - \sqrt{r}}$ by $\frac{3 + \sqrt{r}}{3 + \sqrt{r}}$ .

$\frac{\sqrt{r} (3 + \sqrt{r})}{(3 - \sqrt{r}) (3 + \sqrt{r})}$

Step 3

Expand the denominator using the FOIL method.

$\frac{\sqrt{r} (3 + \sqrt{r})}{9 + 3 \sqrt{r} - 3 \sqrt{r} - {\sqrt{r}}^{2}}$

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

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

$\frac{\sqrt{r} (3 + \sqrt{r})}{9 - r^{\frac{1}{2} \cdot 2}}$

Step 4.3

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

$\frac{\sqrt{r} (3 + \sqrt{r})}{9 - r^{\frac{2}{2}}}$

Step 4.4

Cancel the common factor of $2$ .

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Step 4.4.1

Cancel the common factor.

$\frac{\sqrt{r} (3 + \sqrt{r})}{9 - r^{\frac{2}{2}}}$

Step 4.4.2

Rewrite the expression.

$\frac{\sqrt{r} (3 + \sqrt{r})}{9 - r^{1}}$

Step 4.5

Simplify.

$\frac{\sqrt{r} (3 + \sqrt{r})}{9 - r}$

Step 5

Apply the distributive property.

$\frac{\sqrt{r} \cdot 3 + \sqrt{r} \sqrt{r}}{9 - r}$

Step 6

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

$\frac{3 \cdot \sqrt{r} + \sqrt{r} \sqrt{r}}{9 - r}$

Step 7

Multiply $\sqrt{r} \sqrt{r}$ .

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Step 7.1

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

$\frac{3 \cdot \sqrt{r} + {\sqrt{r}}^{1} \sqrt{r}}{9 - r}$

Step 7.2

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

$\frac{3 \cdot \sqrt{r} + {\sqrt{r}}^{1} {\sqrt{r}}^{1}}{9 - r}$

Step 7.3

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

$\frac{3 \cdot \sqrt{r} + {\sqrt{r}}^{1 + 1}}{9 - r}$

Step 7.4

Add $1$ and $1$ .

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

Cancel the common factor of $2$ .

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Step 8.4.1

Cancel the common factor.

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

Step 8.4.2

Rewrite the expression.

$\frac{3 \sqrt{r} + r^{1}}{9 - r}$

Step 8.5

Simplify.

$\frac{3 \sqrt{r} + r}{9 - r}$

Step 9

Simplify the numerator.

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Step 9.1

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

$\frac{3 r^{\frac{1}{2}} + r}{9 - r}$

Step 9.2

Factor $r^{\frac{1}{2}}$ out of $3 r^{\frac{1}{2}} + r$ .

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Step 9.2.1

Factor $r^{\frac{1}{2}}$ out of $3 r^{\frac{1}{2}}$ .

$\frac{r^{\frac{1}{2}} \cdot 3 + r}{9 - r}$

Step 9.2.2

Raise $r$ to the power of $1$ .

$\frac{r^{\frac{1}{2}} \cdot 3 + r^{1}}{9 - r}$

Step 9.2.3

Factor $r^{\frac{1}{2}}$ out of $r^{1}$ .

$\frac{r^{\frac{1}{2}} \cdot 3 + r^{\frac{1}{2}} r^{\frac{1}{2}}}{9 - r}$

Step 9.2.4

Factor $r^{\frac{1}{2}}$ out of $r^{\frac{1}{2}} \cdot 3 + r^{\frac{1}{2}} r^{\frac{1}{2}}$ .

$\frac{r^{\frac{1}{2}} (3 + r^{\frac{1}{2}})}{9 - r}$