Basic Math Examples

$\frac{8}{r \sqrt{5}} - \frac{\sqrt{5}}{r}$

Step 1

Simplify each term.

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Step 1.1

Multiply $\frac{8}{r \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{8}{r \sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} - \frac{\sqrt{5}}{r}$

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

Multiply $\frac{8}{r \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{8 \sqrt{5}}{r \sqrt{5} \sqrt{5}} - \frac{\sqrt{5}}{r}$

Step 1.2.2

Move $\sqrt{5}$ .

$\frac{8 \sqrt{5}}{r (\sqrt{5} \sqrt{5})} - \frac{\sqrt{5}}{r}$

Step 1.2.3

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

$\frac{8 \sqrt{5}}{r ({\sqrt{5}}^{1} \sqrt{5})} - \frac{\sqrt{5}}{r}$

Step 1.2.4

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

$\frac{8 \sqrt{5}}{r ({\sqrt{5}}^{1} {\sqrt{5}}^{1})} - \frac{\sqrt{5}}{r}$

Step 1.2.5

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

$\frac{8 \sqrt{5}}{r {\sqrt{5}}^{1 + 1}} - \frac{\sqrt{5}}{r}$

Step 1.2.6

Add $1$ and $1$ .

$\frac{8 \sqrt{5}}{r {\sqrt{5}}^{2}} - \frac{\sqrt{5}}{r}$

Step 1.2.7

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

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Step 1.2.7.1

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

$\frac{8 \sqrt{5}}{r {(5^{\frac{1}{2}})}^{2}} - \frac{\sqrt{5}}{r}$

Step 1.2.7.2

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

$\frac{8 \sqrt{5}}{r \cdot 5^{\frac{1}{2} \cdot 2}} - \frac{\sqrt{5}}{r}$

Step 1.2.7.3

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

$\frac{8 \sqrt{5}}{r \cdot 5^{\frac{2}{2}}} - \frac{\sqrt{5}}{r}$

Step 1.2.7.4

Cancel the common factor of $2$ .

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Step 1.2.7.4.1

Cancel the common factor.

$\frac{8 \sqrt{5}}{r \cdot 5^{\frac{2}{2}}} - \frac{\sqrt{5}}{r}$

Step 1.2.7.4.2

Rewrite the expression.

$\frac{8 \sqrt{5}}{r \cdot 5^{1}} - \frac{\sqrt{5}}{r}$

Step 1.2.7.5

Evaluate the exponent.

$\frac{8 \sqrt{5}}{r \cdot 5} - \frac{\sqrt{5}}{r}$

Step 1.3

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

$\frac{8 \sqrt{5}}{5 r} - \frac{\sqrt{5}}{r}$

Step 2

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

$\frac{8 \sqrt{5}}{5 r} - \frac{\sqrt{5}}{r} \cdot \frac{5}{5}$

Step 3

Write each expression with a common denominator of $5 r$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{\sqrt{5}}{r}$ by $\frac{5}{5}$ .

$\frac{8 \sqrt{5}}{5 r} - \frac{\sqrt{5} \cdot 5}{r \cdot 5}$

Step 3.2

Reorder the factors of $r \cdot 5$ .

$\frac{8 \sqrt{5}}{5 r} - \frac{\sqrt{5} \cdot 5}{5 r}$

Step 4

Combine the numerators over the common denominator.

$\frac{8 \sqrt{5} - \sqrt{5} \cdot 5}{5 r}$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $5$ by $- 1$ .

$\frac{8 \sqrt{5} - 5 \sqrt{5}}{5 r}$

Step 5.2

Subtract $5 \sqrt{5}$ from $8 \sqrt{5}$ .

$\frac{3 \sqrt{5}}{5 r}$