Calculus Examples

$\frac{5 \sqrt{3}}{3 \sqrt{3} - 4 \sqrt{6}}$

Step 1

Multiply $\frac{5 \sqrt{3}}{3 \sqrt{3} - 4 \sqrt{6}}$ by $\frac{3 \sqrt{3} + 4 \sqrt{6}}{3 \sqrt{3} + 4 \sqrt{6}}$ .

$\frac{5 \sqrt{3}}{3 \sqrt{3} - 4 \sqrt{6}} \cdot \frac{3 \sqrt{3} + 4 \sqrt{6}}{3 \sqrt{3} + 4 \sqrt{6}}$

Step 2

Multiply $\frac{5 \sqrt{3}}{3 \sqrt{3} - 4 \sqrt{6}}$ by $\frac{3 \sqrt{3} + 4 \sqrt{6}}{3 \sqrt{3} + 4 \sqrt{6}}$ .

$\frac{5 \sqrt{3} (3 \sqrt{3} + 4 \sqrt{6})}{(3 \sqrt{3} - 4 \sqrt{6}) (3 \sqrt{3} + 4 \sqrt{6})}$

Step 3

Expand the denominator using the FOIL method.

$\frac{5 \sqrt{3} (3 \sqrt{3} + 4 \sqrt{6})}{9 {\sqrt{3}}^{2} + 12 \sqrt{18} - 12 \sqrt{18} - 16 {\sqrt{6}}^{2}}$

Step 4

Simplify.

$\frac{5 \sqrt{3} (3 \sqrt{3} + 4 \sqrt{6})}{- 69}$

Step 5

Group $3 \sqrt{3} + 4 \sqrt{6}$ and $\sqrt{3}$ together.

$\frac{5 ((3 \sqrt{3} + 4 \sqrt{6}) \sqrt{3})}{- 69}$

Step 6

Apply the distributive property.

$\frac{5 (3 \sqrt{3} \sqrt{3} + 4 \sqrt{6} \sqrt{3})}{- 69}$

Step 7

Multiply $3 \sqrt{3} \sqrt{3}$ .

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Step 7.1

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

$\frac{5 (3 ({\sqrt{3}}^{1} \sqrt{3}) + 4 \sqrt{6} \sqrt{3})}{- 69}$

Step 7.2

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

$\frac{5 (3 ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) + 4 \sqrt{6} \sqrt{3})}{- 69}$

Step 7.3

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

$\frac{5 (3 {\sqrt{3}}^{1 + 1} + 4 \sqrt{6} \sqrt{3})}{- 69}$

Step 7.4

Add $1$ and $1$ .

$\frac{5 (3 {\sqrt{3}}^{2} + 4 \sqrt{6} \sqrt{3})}{- 69}$

Step 8

Multiply $4 \sqrt{6} \sqrt{3}$ .

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Step 8.1

Combine using the product rule for radicals.

$\frac{5 (3 {\sqrt{3}}^{2} + 4 \sqrt{3 \cdot 6})}{- 69}$

Step 8.2

Multiply $3$ by $6$ .

$\frac{5 (3 {\sqrt{3}}^{2} + 4 \sqrt{18})}{- 69}$

Step 9

Simplify each term.

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Step 9.1

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

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Step 9.1.1

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

$\frac{5 (3 {(3^{\frac{1}{2}})}^{2} + 4 \sqrt{18})}{- 69}$

Step 9.1.2

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

$\frac{5 (3 \cdot 3^{\frac{1}{2} \cdot 2} + 4 \sqrt{18})}{- 69}$

Step 9.1.3

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

$\frac{5 (3 \cdot 3^{\frac{2}{2}} + 4 \sqrt{18})}{- 69}$

Step 9.1.4

Cancel the common factor of $2$ .

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Step 9.1.4.1

Cancel the common factor.

$\frac{5 (3 \cdot 3^{\frac{2}{2}} + 4 \sqrt{18})}{- 69}$

Step 9.1.4.2

Rewrite the expression.

$\frac{5 (3 \cdot 3^{1} + 4 \sqrt{18})}{- 69}$

Step 9.1.5

Evaluate the exponent.

$\frac{5 (3 \cdot 3 + 4 \sqrt{18})}{- 69}$

Step 9.2

Multiply $3$ by $3$ .

$\frac{5 (9 + 4 \sqrt{18})}{- 69}$

Step 9.3

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 9.3.1

Factor $9$ out of $18$ .

$\frac{5 (9 + 4 \sqrt{9 (2)})}{- 69}$

Step 9.3.2

Rewrite $9$ as $3^{2}$ .

$\frac{5 (9 + 4 \sqrt{3^{2} \cdot 2})}{- 69}$

Step 9.4

Pull terms out from under the radical.

$\frac{5 (9 + 4 (3 \sqrt{2}))}{- 69}$

Step 9.5

Multiply $3$ by $4$ .

$\frac{5 (9 + 12 \sqrt{2})}{- 69}$

Step 10

Reduce the expression by cancelling the common factors.

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Step 10.1

Cancel the common factor of $9 + 12 \sqrt{2}$ and $- 69$ .

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Step 10.1.1

Factor $3$ out of $5 (9 + 12 \sqrt{2})$ .

$\frac{3 (5 (3 + 4 \sqrt{2}))}{- 69}$

Step 10.1.2

Cancel the common factors.

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Step 10.1.2.1

Factor $3$ out of $- 69$ .

$\frac{3 (5 (3 + 4 \sqrt{2}))}{3 (- 23)}$

Step 10.1.2.2

Cancel the common factor.

$\frac{3 (5 (3 + 4 \sqrt{2}))}{3 \cdot - 23}$

Step 10.1.2.3

Rewrite the expression.

$\frac{5 (3 + 4 \sqrt{2})}{- 23}$

Step 10.2

Move the negative in front of the fraction.

$- \frac{5 (3 + 4 \sqrt{2})}{23}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- \frac{5 (3 + 4 \sqrt{2})}{23}$

Decimal Form:

$- 1.88192483 \dots$