Calculus Examples

$\sqrt{6} - \frac{5}{\sqrt{6}} + \frac{22}{{(\sqrt{6})}^{3}}$

Step 1

Simplify each term.

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Step 1.1

Multiply $\frac{5}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\sqrt{6} - (\frac{5}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}) + \frac{22}{{(\sqrt{6})}^{3}}$

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

Multiply $\frac{5}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\sqrt{6} - \frac{5 \sqrt{6}}{\sqrt{6} \sqrt{6}} + \frac{22}{{(\sqrt{6})}^{3}}$

Step 1.2.2

Raise $\sqrt{6}$ to the power of $1$ .

$\sqrt{6} - \frac{5 \sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}} + \frac{22}{{(\sqrt{6})}^{3}}$

Step 1.2.3

Raise $\sqrt{6}$ to the power of $1$ .

$\sqrt{6} - \frac{5 \sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}} + \frac{22}{{(\sqrt{6})}^{3}}$

Step 1.2.4

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

$\sqrt{6} - \frac{5 \sqrt{6}}{{\sqrt{6}}^{1 + 1}} + \frac{22}{{(\sqrt{6})}^{3}}$

Step 1.2.5

Add $1$ and $1$ .

$\sqrt{6} - \frac{5 \sqrt{6}}{{\sqrt{6}}^{2}} + \frac{22}{{(\sqrt{6})}^{3}}$

Step 1.2.6

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 1.2.6.1

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

$\sqrt{6} - \frac{5 \sqrt{6}}{{(6^{\frac{1}{2}})}^{2}} + \frac{22}{{(\sqrt{6})}^{3}}$

Step 1.2.6.2

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

$\sqrt{6} - \frac{5 \sqrt{6}}{6^{\frac{1}{2} \cdot 2}} + \frac{22}{{(\sqrt{6})}^{3}}$

Step 1.2.6.3

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

$\sqrt{6} - \frac{5 \sqrt{6}}{6^{\frac{2}{2}}} + \frac{22}{{(\sqrt{6})}^{3}}$

Step 1.2.6.4

Cancel the common factor of $2$ .

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Step 1.2.6.4.1

Cancel the common factor.

$\sqrt{6} - \frac{5 \sqrt{6}}{6^{\frac{2}{2}}} + \frac{22}{{(\sqrt{6})}^{3}}$

Step 1.2.6.4.2

Rewrite the expression.

$\sqrt{6} - \frac{5 \sqrt{6}}{6^{1}} + \frac{22}{{(\sqrt{6})}^{3}}$

Step 1.2.6.5

Evaluate the exponent.

$\sqrt{6} - \frac{5 \sqrt{6}}{6} + \frac{22}{{(\sqrt{6})}^{3}}$

Step 1.3

Simplify the denominator.

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Step 1.3.1

Rewrite ${\sqrt{6}}^{3}$ as $\sqrt{6^{3}}$ .

$\sqrt{6} - \frac{5 \sqrt{6}}{6} + \frac{22}{\sqrt{6^{3}}}$

Step 1.3.2

Raise $6$ to the power of $3$ .

$\sqrt{6} - \frac{5 \sqrt{6}}{6} + \frac{22}{\sqrt{216}}$

Step 1.3.3

Rewrite $216$ as $6^{2} \cdot 6$ .

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Step 1.3.3.1

Factor $36$ out of $216$ .

$\sqrt{6} - \frac{5 \sqrt{6}}{6} + \frac{22}{\sqrt{36 (6)}}$

Step 1.3.3.2

Rewrite $36$ as $6^{2}$ .

$\sqrt{6} - \frac{5 \sqrt{6}}{6} + \frac{22}{\sqrt{6^{2} \cdot 6}}$

Step 1.3.4

Pull terms out from under the radical.

$\sqrt{6} - \frac{5 \sqrt{6}}{6} + \frac{22}{6 \sqrt{6}}$

Step 1.4

Cancel the common factor of $22$ and $6$ .

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Step 1.4.1

Factor $2$ out of $22$ .

$\sqrt{6} - \frac{5 \sqrt{6}}{6} + \frac{2 (11)}{6 \sqrt{6}}$

Step 1.4.2

Cancel the common factors.

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Step 1.4.2.1

Factor $2$ out of $6 \sqrt{6}$ .

$\sqrt{6} - \frac{5 \sqrt{6}}{6} + \frac{2 (11)}{2 (3 \sqrt{6})}$

Step 1.4.2.2

Cancel the common factor.

$\sqrt{6} - \frac{5 \sqrt{6}}{6} + \frac{2 \cdot 11}{2 (3 \sqrt{6})}$

Step 1.4.2.3

Rewrite the expression.

$\sqrt{6} - \frac{5 \sqrt{6}}{6} + \frac{11}{3 \sqrt{6}}$

Step 1.5

Multiply $\frac{11}{3 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\sqrt{6} - \frac{5 \sqrt{6}}{6} + \frac{11}{3 \sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Step 1.6

Combine and simplify the denominator.

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Step 1.6.1

Multiply $\frac{11}{3 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\sqrt{6} - \frac{5 \sqrt{6}}{6} + \frac{11 \sqrt{6}}{3 \sqrt{6} \sqrt{6}}$

Step 1.6.2

Move $\sqrt{6}$ .

$\sqrt{6} - \frac{5 \sqrt{6}}{6} + \frac{11 \sqrt{6}}{3 (\sqrt{6} \sqrt{6})}$

Step 1.6.3

Raise $\sqrt{6}$ to the power of $1$ .

$\sqrt{6} - \frac{5 \sqrt{6}}{6} + \frac{11 \sqrt{6}}{3 ({\sqrt{6}}^{1} \sqrt{6})}$

Step 1.6.4

Raise $\sqrt{6}$ to the power of $1$ .

$\sqrt{6} - \frac{5 \sqrt{6}}{6} + \frac{11 \sqrt{6}}{3 ({\sqrt{6}}^{1} {\sqrt{6}}^{1})}$

Step 1.6.5

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

$\sqrt{6} - \frac{5 \sqrt{6}}{6} + \frac{11 \sqrt{6}}{3 {\sqrt{6}}^{1 + 1}}$

Step 1.6.6

Add $1$ and $1$ .

$\sqrt{6} - \frac{5 \sqrt{6}}{6} + \frac{11 \sqrt{6}}{3 {\sqrt{6}}^{2}}$

Step 1.6.7

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 1.6.7.1

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

$\sqrt{6} - \frac{5 \sqrt{6}}{6} + \frac{11 \sqrt{6}}{3 {(6^{\frac{1}{2}})}^{2}}$

Step 1.6.7.2

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

$\sqrt{6} - \frac{5 \sqrt{6}}{6} + \frac{11 \sqrt{6}}{3 \cdot 6^{\frac{1}{2} \cdot 2}}$

Step 1.6.7.3

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

$\sqrt{6} - \frac{5 \sqrt{6}}{6} + \frac{11 \sqrt{6}}{3 \cdot 6^{\frac{2}{2}}}$

Step 1.6.7.4

Cancel the common factor of $2$ .

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Step 1.6.7.4.1

Cancel the common factor.

$\sqrt{6} - \frac{5 \sqrt{6}}{6} + \frac{11 \sqrt{6}}{3 \cdot 6^{\frac{2}{2}}}$

Step 1.6.7.4.2

Rewrite the expression.

$\sqrt{6} - \frac{5 \sqrt{6}}{6} + \frac{11 \sqrt{6}}{3 \cdot 6^{1}}$

Step 1.6.7.5

Evaluate the exponent.

$\sqrt{6} - \frac{5 \sqrt{6}}{6} + \frac{11 \sqrt{6}}{3 \cdot 6}$

Step 1.7

Multiply $3$ by $6$ .

$\sqrt{6} - \frac{5 \sqrt{6}}{6} + \frac{11 \sqrt{6}}{18}$

Step 2

Find the common denominator.

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Step 2.1

Write $\sqrt{6}$ as a fraction with denominator $1$ .

$\frac{\sqrt{6}}{1} - \frac{5 \sqrt{6}}{6} + \frac{11 \sqrt{6}}{18}$

Step 2.2

Multiply $\frac{\sqrt{6}}{1}$ by $\frac{18}{18}$ .

$\frac{\sqrt{6}}{1} \cdot \frac{18}{18} - \frac{5 \sqrt{6}}{6} + \frac{11 \sqrt{6}}{18}$

Step 2.3

Multiply $\frac{\sqrt{6}}{1}$ by $\frac{18}{18}$ .

$\frac{\sqrt{6} \cdot 18}{18} - \frac{5 \sqrt{6}}{6} + \frac{11 \sqrt{6}}{18}$

Step 2.4

Multiply $\frac{5 \sqrt{6}}{6}$ by $\frac{3}{3}$ .

$\frac{\sqrt{6} \cdot 18}{18} - (\frac{5 \sqrt{6}}{6} \cdot \frac{3}{3}) + \frac{11 \sqrt{6}}{18}$

Step 2.5

Multiply $\frac{5 \sqrt{6}}{6}$ by $\frac{3}{3}$ .

$\frac{\sqrt{6} \cdot 18}{18} - \frac{5 \sqrt{6} \cdot 3}{6 \cdot 3} + \frac{11 \sqrt{6}}{18}$

Step 2.6

Reorder the factors of $6 \cdot 3$ .

$\frac{\sqrt{6} \cdot 18}{18} - \frac{5 \sqrt{6} \cdot 3}{3 \cdot 6} + \frac{11 \sqrt{6}}{18}$

Step 2.7

Multiply $3$ by $6$ .

$\frac{\sqrt{6} \cdot 18}{18} - \frac{5 \sqrt{6} \cdot 3}{18} + \frac{11 \sqrt{6}}{18}$

Step 3

Combine the numerators over the common denominator.

$\frac{\sqrt{6} \cdot 18 - 5 \sqrt{6} \cdot 3 + 11 \sqrt{6}}{18}$

Step 4

Simplify each term.

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Step 4.1

Move $18$ to the left of $\sqrt{6}$ .

$\frac{18 \cdot \sqrt{6} - 5 \sqrt{6} \cdot 3 + 11 \sqrt{6}}{18}$

Step 4.2

Multiply $3$ by $- 5$ .

$\frac{18 \sqrt{6} - 15 \sqrt{6} + 11 \sqrt{6}}{18}$

Step 5

Simplify terms.

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Step 5.1

Subtract $15 \sqrt{6}$ from $18 \sqrt{6}$ .

$\frac{3 \sqrt{6} + 11 \sqrt{6}}{18}$

Step 5.2

Add $3 \sqrt{6}$ and $11 \sqrt{6}$ .

$\frac{14 \sqrt{6}}{18}$

Step 5.3

Cancel the common factor of $14$ and $18$ .

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Step 5.3.1

Factor $2$ out of $14 \sqrt{6}$ .

$\frac{2 (7 \sqrt{6})}{18}$

Step 5.3.2

Cancel the common factors.

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Step 5.3.2.1

Factor $2$ out of $18$ .

$\frac{2 (7 \sqrt{6})}{2 (9)}$

Step 5.3.2.2

Cancel the common factor.

$\frac{2 (7 \sqrt{6})}{2 \cdot 9}$

Step 5.3.2.3

Rewrite the expression.

$\frac{7 \sqrt{6}}{9}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{7 \sqrt{6}}{9}$

Decimal Form:

$1.90515868 \dots$