Algebra Examples

$\frac{- 2 \sqrt{7} + 5 \sqrt{2}}{\sqrt{3 r}}$

Step 1

Multiply $\frac{- 2 \sqrt{7} + 5 \sqrt{2}}{\sqrt{3 r}}$ by $\frac{\sqrt{3 r}}{\sqrt{3 r}}$ .

$\frac{- 2 \sqrt{7} + 5 \sqrt{2}}{\sqrt{3 r}} \cdot \frac{\sqrt{3 r}}{\sqrt{3 r}}$

Step 2

Combine and simplify the denominator.

Tap for more steps...

Step 2.1

Multiply $\frac{- 2 \sqrt{7} + 5 \sqrt{2}}{\sqrt{3 r}}$ by $\frac{\sqrt{3 r}}{\sqrt{3 r}}$ .

$\frac{(- 2 \sqrt{7} + 5 \sqrt{2}) \sqrt{3 r}}{\sqrt{3 r} \sqrt{3 r}}$

Step 2.2

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

$\frac{(- 2 \sqrt{7} + 5 \sqrt{2}) \sqrt{3 r}}{{\sqrt{3 r}}^{1} \sqrt{3 r}}$

Step 2.3

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

$\frac{(- 2 \sqrt{7} + 5 \sqrt{2}) \sqrt{3 r}}{{\sqrt{3 r}}^{1} {\sqrt{3 r}}^{1}}$

Step 2.4

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

$\frac{(- 2 \sqrt{7} + 5 \sqrt{2}) \sqrt{3 r}}{{\sqrt{3 r}}^{1 + 1}}$

Step 2.5

Add $1$ and $1$ .

$\frac{(- 2 \sqrt{7} + 5 \sqrt{2}) \sqrt{3 r}}{{\sqrt{3 r}}^{2}}$

Step 2.6

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

Tap for more steps...

Step 2.6.1

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

$\frac{(- 2 \sqrt{7} + 5 \sqrt{2}) \sqrt{3 r}}{{({(3 r)}^{\frac{1}{2}})}^{2}}$

Step 2.6.2

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

$\frac{(- 2 \sqrt{7} + 5 \sqrt{2}) \sqrt{3 r}}{{(3 r)}^{\frac{1}{2} \cdot 2}}$

Step 2.6.3

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

$\frac{(- 2 \sqrt{7} + 5 \sqrt{2}) \sqrt{3 r}}{{(3 r)}^{\frac{2}{2}}}$

Step 2.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.6.4.1

Cancel the common factor.

$\frac{(- 2 \sqrt{7} + 5 \sqrt{2}) \sqrt{3 r}}{{(3 r)}^{\frac{2}{2}}}$

Step 2.6.4.2

Rewrite the expression.

$\frac{(- 2 \sqrt{7} + 5 \sqrt{2}) \sqrt{3 r}}{{(3 r)}^{1}}$

Step 2.6.5

Simplify.

$\frac{(- 2 \sqrt{7} + 5 \sqrt{2}) \sqrt{3 r}}{3 r}$

Step 3

Apply the distributive property.

$\frac{- 2 \sqrt{7} \sqrt{3 r} + 5 \sqrt{2} \sqrt{3 r}}{3 r}$

Step 4

Multiply $- 2 \sqrt{7} \sqrt{3 r}$ .

Tap for more steps...

Step 4.1

Combine using the product rule for radicals.

$\frac{- 2 \sqrt{3 r \cdot 7} + 5 \sqrt{2} \sqrt{3 r}}{3 r}$

Step 4.2

Multiply $7$ by $3$ .

$\frac{- 2 \sqrt{21 r} + 5 \sqrt{2} \sqrt{3 r}}{3 r}$

Step 5

Multiply $5 \sqrt{2} \sqrt{3 r}$ .

Tap for more steps...

Step 5.1

Combine using the product rule for radicals.

$\frac{- 2 \sqrt{21 r} + 5 \sqrt{3 r \cdot 2}}{3 r}$

Step 5.2

Multiply $2$ by $3$ .

$\frac{- 2 \sqrt{21 r} + 5 \sqrt{6 r}}{3 r}$

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21 r}$ as ${(21 r)}^{\frac{1}{2}}$ .

$\frac{- 2 {(21 r)}^{\frac{1}{2}} + 5 \sqrt{6 r}}{3 r}$

Step 6.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 r}$ as ${(6 r)}^{\frac{1}{2}}$ .

$\frac{- 2 {(21 r)}^{\frac{1}{2}} + 5 {(6 r)}^{\frac{1}{2}}}{3 r}$

Step 6.3

Apply the product rule to $21 r$ .

$\frac{- 2 (21^{\frac{1}{2}} r^{\frac{1}{2}}) + 5 {(6 r)}^{\frac{1}{2}}}{3 r}$

Step 6.4

Apply the product rule to $6 r$ .

$\frac{- 2 \cdot 21^{\frac{1}{2}} r^{\frac{1}{2}} + 5 (6^{\frac{1}{2}} r^{\frac{1}{2}})}{3 r}$

Step 6.5

Factor $r^{\frac{1}{2}}$ out of $- 2 \cdot 21^{\frac{1}{2}} r^{\frac{1}{2}} + 5 \cdot 6^{\frac{1}{2}} r^{\frac{1}{2}}$ .

Tap for more steps...

Step 6.5.1

Factor $r^{\frac{1}{2}}$ out of $- 2 \cdot 21^{\frac{1}{2}} r^{\frac{1}{2}}$ .

$\frac{r^{\frac{1}{2}} (- 2 \cdot 21^{\frac{1}{2}}) + 5 \cdot 6^{\frac{1}{2}} r^{\frac{1}{2}}}{3 r}$

Step 6.5.2

Factor $r^{\frac{1}{2}}$ out of $5 \cdot 6^{\frac{1}{2}} r^{\frac{1}{2}}$ .

$\frac{r^{\frac{1}{2}} (- 2 \cdot 21^{\frac{1}{2}}) + r^{\frac{1}{2}} (5 \cdot 6^{\frac{1}{2}})}{3 r}$

Step 6.5.3

Factor $r^{\frac{1}{2}}$ out of $r^{\frac{1}{2}} (- 2 \cdot 21^{\frac{1}{2}}) + r^{\frac{1}{2}} (5 \cdot 6^{\frac{1}{2}})$ .

$\frac{r^{\frac{1}{2}} (- 2 \cdot 21^{\frac{1}{2}} + 5 \cdot 6^{\frac{1}{2}})}{3 r}$

Step 7

Move $r^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{- 2 \cdot 21^{\frac{1}{2}} + 5 \cdot 6^{\frac{1}{2}}}{3 r \cdot r^{- \frac{1}{2}}}$

Step 8

Multiply $r$ by $r^{- \frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 8.1

Move $r^{- \frac{1}{2}}$ .

$\frac{- 2 \cdot 21^{\frac{1}{2}} + 5 \cdot 6^{\frac{1}{2}}}{3 (r^{- \frac{1}{2}} r)}$

Step 8.2

Multiply $r^{- \frac{1}{2}}$ by $r$ .

Tap for more steps...

Step 8.2.1

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

$\frac{- 2 \cdot 21^{\frac{1}{2}} + 5 \cdot 6^{\frac{1}{2}}}{3 (r^{- \frac{1}{2}} r^{1})}$

Step 8.2.2

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

$\frac{- 2 \cdot 21^{\frac{1}{2}} + 5 \cdot 6^{\frac{1}{2}}}{3 r^{- \frac{1}{2} + 1}}$

Step 8.3

Write $1$ as a fraction with a common denominator.

$\frac{- 2 \cdot 21^{\frac{1}{2}} + 5 \cdot 6^{\frac{1}{2}}}{3 r^{- \frac{1}{2} + \frac{2}{2}}}$

Step 8.4

Combine the numerators over the common denominator.

$\frac{- 2 \cdot 21^{\frac{1}{2}} + 5 \cdot 6^{\frac{1}{2}}}{3 r^{\frac{- 1 + 2}{2}}}$

Step 8.5

Add $- 1$ and $2$ .

$\frac{- 2 \cdot 21^{\frac{1}{2}} + 5 \cdot 6^{\frac{1}{2}}}{3 r^{\frac{1}{2}}}$

Step 9

Factor $- 1$ out of $- 2 \cdot 21^{\frac{1}{2}}$ .

$\frac{- (2 \cdot 21^{\frac{1}{2}}) + 5 \cdot 6^{\frac{1}{2}}}{3 r^{\frac{1}{2}}}$

Step 10

Factor $- 1$ out of $5 \cdot 6^{\frac{1}{2}}$ .

$\frac{- (2 \cdot 21^{\frac{1}{2}}) - (- 5 \cdot 6^{\frac{1}{2}})}{3 r^{\frac{1}{2}}}$

Step 11

Factor $- 1$ out of $- (2 \cdot 21^{\frac{1}{2}}) - (- 5 \cdot 6^{\frac{1}{2}})$ .

$\frac{- (2 \cdot 21^{\frac{1}{2}} - 5 \cdot 6^{\frac{1}{2}})}{3 r^{\frac{1}{2}}}$

Step 12

Simplify the expression.

Tap for more steps...

Step 12.1

Rewrite $- (2 \cdot 21^{\frac{1}{2}} - 5 \cdot 6^{\frac{1}{2}})$ as $- 1 (2 \cdot 21^{\frac{1}{2}} - 5 \cdot 6^{\frac{1}{2}})$ .

$\frac{- 1 (2 \cdot 21^{\frac{1}{2}} - 5 \cdot 6^{\frac{1}{2}})}{3 r^{\frac{1}{2}}}$

Step 12.2

Move the negative in front of the fraction.

$- \frac{2 \cdot 21^{\frac{1}{2}} - 5 \cdot 6^{\frac{1}{2}}}{3 r^{\frac{1}{2}}}$