Trigonometry Examples

$\csc (15)$

Step 1

Split $15$ into two angles where the values of the six trigonometric functions are known.

$\csc (45 - 30)$

Step 2

Separate negation.

$\csc (45 - (30))$

Step 3

Apply the difference of angles identity.

$\frac{\sec (45) \sec (30) \csc (45) \csc (30)}{\sec (45) \csc (30) - \csc (45) \sec (30)}$

Step 4

The exact value of $\sec (45)$ is $\frac{2}{\sqrt{2}}$ .

$\frac{\frac{2}{\sqrt{2}} \sec (30) \csc (45) \csc (30)}{\sec (45) \csc (30) - \csc (45) \sec (30)}$

Step 5

The exact value of $\sec (30)$ is $\frac{2}{\sqrt{3}}$ .

$\frac{\frac{2}{\sqrt{2}} \cdot \frac{2}{\sqrt{3}} \csc (45) \csc (30)}{\sec (45) \csc (30) - \csc (45) \sec (30)}$

Step 6

The exact value of $\csc (45)$ is $\sqrt{2}$ .

$\frac{\frac{2}{\sqrt{2}} \cdot \frac{2}{\sqrt{3}} \sqrt{2} \csc (30)}{\sec (45) \csc (30) - \csc (45) \sec (30)}$

Step 7

The exact value of $\csc (30)$ is $2$ .

$\frac{\frac{2}{\sqrt{2}} \cdot \frac{2}{\sqrt{3}} \sqrt{2} \cdot 2}{\sec (45) \csc (30) - \csc (45) \sec (30)}$

Step 8

The exact value of $\sec (45)$ is $\frac{2}{\sqrt{2}}$ .

$\frac{\frac{2}{\sqrt{2}} \cdot \frac{2}{\sqrt{3}} \sqrt{2} \cdot 2}{\frac{2}{\sqrt{2}} \csc (30) - \csc (45) \sec (30)}$

Step 9

The exact value of $\csc (30)$ is $2$ .

$\frac{\frac{2}{\sqrt{2}} \cdot \frac{2}{\sqrt{3}} \sqrt{2} \cdot 2}{\frac{2}{\sqrt{2}} \cdot 2 - \csc (45) \sec (30)}$

Step 10

The exact value of $\csc (45)$ is $\sqrt{2}$ .

$\frac{\frac{2}{\sqrt{2}} \cdot \frac{2}{\sqrt{3}} \sqrt{2} \cdot 2}{\frac{2}{\sqrt{2}} \cdot 2 - \sqrt{2} \sec (30)}$

Step 11

The exact value of $\sec (30)$ is $\frac{2}{\sqrt{3}}$ .

$\frac{\frac{2}{\sqrt{2}} \cdot \frac{2}{\sqrt{3}} \sqrt{2} \cdot 2}{\frac{2}{\sqrt{2}} \cdot 2 - \sqrt{2} \frac{2}{\sqrt{3}}}$

Step 12

Simplify $\frac{\frac{2}{\sqrt{2}} \cdot \frac{2}{\sqrt{3}} \sqrt{2} \cdot 2}{\frac{2}{\sqrt{2}} \cdot 2 - \sqrt{2} \frac{2}{\sqrt{3}}}$ .

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Step 12.1

Simplify the numerator.

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Step 12.1.1

Multiply $\frac{2}{\sqrt{2}}$ by $\frac{2}{\sqrt{3}}$ .

$\frac{\frac{2 \cdot 2}{\sqrt{2} \sqrt{3}} \sqrt{2} \cdot 2}{\frac{2}{\sqrt{2}} \cdot 2 - \sqrt{2} \frac{2}{\sqrt{3}}}$

Step 12.1.2

Combine $\frac{2 \cdot 2}{\sqrt{2} \sqrt{3}}$ and $\sqrt{2}$ .

$\frac{\frac{2 \cdot 2 \sqrt{2}}{\sqrt{2} \sqrt{3}} \cdot 2}{\frac{2}{\sqrt{2}} \cdot 2 - \sqrt{2} \frac{2}{\sqrt{3}}}$

Step 12.1.3

Combine $\frac{2 \cdot 2 \sqrt{2}}{\sqrt{2} \sqrt{3}}$ and $2$ .

$\frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{\frac{2}{\sqrt{2}} \cdot 2 - \sqrt{2} \frac{2}{\sqrt{3}}}$

Step 12.2

Simplify the denominator.

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Step 12.2.1

Multiply $\frac{2}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} \cdot 2 - \sqrt{2} \frac{2}{\sqrt{3}}}$

Step 12.2.2

Combine and simplify the denominator.

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Step 12.2.2.1

Multiply $\frac{2}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{\frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}} \cdot 2 - \sqrt{2} \frac{2}{\sqrt{3}}}$

Step 12.2.2.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 12.2.2.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 12.2.2.4

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

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

Step 12.2.2.5

Add $1$ and $1$ .

$\frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{\frac{2 \sqrt{2}}{{\sqrt{2}}^{2}} \cdot 2 - \sqrt{2} \frac{2}{\sqrt{3}}}$

Step 12.2.2.6

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

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Step 12.2.2.6.1

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

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

Step 12.2.2.6.2

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

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

Step 12.2.2.6.3

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

$\frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{\frac{2 \sqrt{2}}{2^{\frac{2}{2}}} \cdot 2 - \sqrt{2} \frac{2}{\sqrt{3}}}$

Step 12.2.2.6.4

Cancel the common factor of $2$ .

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Step 12.2.2.6.4.1

Cancel the common factor.

$\frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{\frac{2 \sqrt{2}}{2^{\frac{2}{2}}} \cdot 2 - \sqrt{2} \frac{2}{\sqrt{3}}}$

Step 12.2.2.6.4.2

Rewrite the expression.

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

Step 12.2.2.6.5

Evaluate the exponent.

$\frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{\frac{2 \sqrt{2}}{2} \cdot 2 - \sqrt{2} \frac{2}{\sqrt{3}}}$

Step 12.2.3

Cancel the common factor of $2$ .

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Step 12.2.3.1

Cancel the common factor.

$\frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{\frac{2 \sqrt{2}}{2} \cdot 2 - \sqrt{2} \frac{2}{\sqrt{3}}}$

Step 12.2.3.2

Rewrite the expression.

$\frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{2 \sqrt{2} - \sqrt{2} \frac{2}{\sqrt{3}}}$

Step 12.2.4

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{2 \sqrt{2} - \sqrt{2} (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})}$

Step 12.2.5

Combine and simplify the denominator.

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Step 12.2.5.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{2 \sqrt{2} - \sqrt{2} \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}}$

Step 12.2.5.2

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

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

Step 12.2.5.3

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

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

Step 12.2.5.4

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

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

Step 12.2.5.5

Add $1$ and $1$ .

$\frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{2 \sqrt{2} - \sqrt{2} \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}}}$

Step 12.2.5.6

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

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Step 12.2.5.6.1

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

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

Step 12.2.5.6.2

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

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

Step 12.2.5.6.3

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

$\frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{2 \sqrt{2} - \sqrt{2} \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}}$

Step 12.2.5.6.4

Cancel the common factor of $2$ .

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Step 12.2.5.6.4.1

Cancel the common factor.

$\frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{2 \sqrt{2} - \sqrt{2} \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}}$

Step 12.2.5.6.4.2

Rewrite the expression.

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

Step 12.2.5.6.5

Evaluate the exponent.

$\frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{2 \sqrt{2} - \sqrt{2} \frac{2 \sqrt{3}}{3}}$

Step 12.2.6

Multiply $- \sqrt{2} \frac{2 \sqrt{3}}{3}$ .

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Step 12.2.6.1

Combine $\frac{2 \sqrt{3}}{3}$ and $\sqrt{2}$ .

$\frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{2 \sqrt{2} - \frac{2 \sqrt{3} \sqrt{2}}{3}}$

Step 12.2.6.2

Combine using the product rule for radicals.

$\frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{2 \sqrt{2} - \frac{2 \sqrt{2 \cdot 3}}{3}}$

Step 12.2.6.3

Multiply $2$ by $3$ .

$\frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{2 \sqrt{2} - \frac{2 \sqrt{6}}{3}}$

Step 12.2.7

To write $2 \sqrt{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{2 \sqrt{2} \cdot \frac{3}{3} - \frac{2 \sqrt{6}}{3}}$

Step 12.2.8

Combine $2 \sqrt{2}$ and $\frac{3}{3}$ .

$\frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{\frac{2 \sqrt{2} \cdot 3}{3} - \frac{2 \sqrt{6}}{3}}$

Step 12.2.9

Combine the numerators over the common denominator.

$\frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{\frac{2 \sqrt{2} \cdot 3 - 2 \sqrt{6}}{3}}$

Step 12.2.10

Multiply $3$ by $2$ .

$\frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{\frac{6 \sqrt{2} - 2 \sqrt{6}}{3}}$

Step 12.3

Simplify the numerator.

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Step 12.3.1

Multiply $2$ by $2$ .

$\frac{\frac{4 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{\frac{6 \sqrt{2} - 2 \sqrt{6}}{3}}$

Step 12.3.2

Multiply $2$ by $4$ .

$\frac{\frac{8 \sqrt{2}}{\sqrt{2} \sqrt{3}}}{\frac{6 \sqrt{2} - 2 \sqrt{6}}{3}}$

Step 12.4

Simplify the denominator.

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Step 12.4.1

Combine using the product rule for radicals.

$\frac{\frac{8 \sqrt{2}}{\sqrt{2 \cdot 3}}}{\frac{6 \sqrt{2} - 2 \sqrt{6}}{3}}$

Step 12.4.2

Multiply $2$ by $3$ .

$\frac{\frac{8 \sqrt{2}}{\sqrt{6}}}{\frac{6 \sqrt{2} - 2 \sqrt{6}}{3}}$

Step 12.5

Simplify the numerator.

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Step 12.5.1

Combine $\sqrt{2}$ and $\sqrt{6}$ into a single radical.

$\frac{8 \sqrt{\frac{2}{6}}}{\frac{6 \sqrt{2} - 2 \sqrt{6}}{3}}$

Step 12.5.2

Cancel the common factor of $2$ and $6$ .

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Step 12.5.2.1

Factor $2$ out of $2$ .

$\frac{8 \sqrt{\frac{2 (1)}{6}}}{\frac{6 \sqrt{2} - 2 \sqrt{6}}{3}}$

Step 12.5.2.2

Cancel the common factors.

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Step 12.5.2.2.1

Factor $2$ out of $6$ .

$\frac{8 \sqrt{\frac{2 \cdot 1}{2 \cdot 3}}}{\frac{6 \sqrt{2} - 2 \sqrt{6}}{3}}$

Step 12.5.2.2.2

Cancel the common factor.

$\frac{8 \sqrt{\frac{2 \cdot 1}{2 \cdot 3}}}{\frac{6 \sqrt{2} - 2 \sqrt{6}}{3}}$

Step 12.5.2.2.3

Rewrite the expression.

$\frac{8 \sqrt{\frac{1}{3}}}{\frac{6 \sqrt{2} - 2 \sqrt{6}}{3}}$

Step 12.5.3

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

$\frac{8 \frac{\sqrt{1}}{\sqrt{3}}}{\frac{6 \sqrt{2} - 2 \sqrt{6}}{3}}$

Step 12.5.4

Any root of $1$ is $1$ .

$\frac{8 \frac{1}{\sqrt{3}}}{\frac{6 \sqrt{2} - 2 \sqrt{6}}{3}}$

Step 12.5.5

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{8 (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})}{\frac{6 \sqrt{2} - 2 \sqrt{6}}{3}}$

Step 12.5.6

Combine and simplify the denominator.

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Step 12.5.6.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{8 \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}}{\frac{6 \sqrt{2} - 2 \sqrt{6}}{3}}$

Step 12.5.6.2

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

$\frac{8 \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}}{\frac{6 \sqrt{2} - 2 \sqrt{6}}{3}}$

Step 12.5.6.3

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

$\frac{8 \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}}{\frac{6 \sqrt{2} - 2 \sqrt{6}}{3}}$

Step 12.5.6.4

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

$\frac{8 \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}}{\frac{6 \sqrt{2} - 2 \sqrt{6}}{3}}$

Step 12.5.6.5

Add $1$ and $1$ .

$\frac{8 \frac{\sqrt{3}}{{\sqrt{3}}^{2}}}{\frac{6 \sqrt{2} - 2 \sqrt{6}}{3}}$

Step 12.5.6.6

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

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Step 12.5.6.6.1

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

$\frac{8 \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}}{\frac{6 \sqrt{2} - 2 \sqrt{6}}{3}}$

Step 12.5.6.6.2

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

$\frac{8 \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}}{\frac{6 \sqrt{2} - 2 \sqrt{6}}{3}}$

Step 12.5.6.6.3

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

$\frac{8 \frac{\sqrt{3}}{3^{\frac{2}{2}}}}{\frac{6 \sqrt{2} - 2 \sqrt{6}}{3}}$

Step 12.5.6.6.4

Cancel the common factor of $2$ .

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Step 12.5.6.6.4.1

Cancel the common factor.

$\frac{8 \frac{\sqrt{3}}{3^{\frac{2}{2}}}}{\frac{6 \sqrt{2} - 2 \sqrt{6}}{3}}$

Step 12.5.6.6.4.2

Rewrite the expression.

$\frac{8 \frac{\sqrt{3}}{3^{1}}}{\frac{6 \sqrt{2} - 2 \sqrt{6}}{3}}$

Step 12.5.6.6.5

Evaluate the exponent.

$\frac{8 \frac{\sqrt{3}}{3}}{\frac{6 \sqrt{2} - 2 \sqrt{6}}{3}}$

Step 12.5.7

Combine $8$ and $\frac{\sqrt{3}}{3}$ .

$\frac{\frac{8 \sqrt{3}}{3}}{\frac{6 \sqrt{2} - 2 \sqrt{6}}{3}}$

Step 12.6

Multiply the numerator by the reciprocal of the denominator.

$\frac{8 \sqrt{3}}{3} \cdot \frac{3}{6 \sqrt{2} - 2 \sqrt{6}}$

Step 12.7

Cancel the common factor of $3$ .

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Step 12.7.1

Cancel the common factor.

$\frac{8 \sqrt{3}}{3} \cdot \frac{3}{6 \sqrt{2} - 2 \sqrt{6}}$

Step 12.7.2

Rewrite the expression.

$8 \sqrt{3} \frac{1}{6 \sqrt{2} - 2 \sqrt{6}}$

Step 12.8

Combine $\frac{1}{6 \sqrt{2} - 2 \sqrt{6}}$ and $8$ .

$\frac{8}{6 \sqrt{2} - 2 \sqrt{6}} \sqrt{3}$

Step 12.9

Combine $\frac{8}{6 \sqrt{2} - 2 \sqrt{6}}$ and $\sqrt{3}$ .

$\frac{8 \sqrt{3}}{6 \sqrt{2} - 2 \sqrt{6}}$

Step 12.10

Cancel the common factor of $8$ and $6 \sqrt{2} - 2 \sqrt{6}$ .

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Step 12.10.1

Factor $2$ out of $8 \sqrt{3}$ .

$\frac{2 (4 \sqrt{3})}{6 \sqrt{2} - 2 \sqrt{6}}$

Step 12.10.2

Cancel the common factors.

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Step 12.10.2.1

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

$\frac{2 (4 \sqrt{3})}{2 (3 \sqrt{2}) - 2 \sqrt{6}}$

Step 12.10.2.2

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

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

Step 12.10.2.3

Factor $2$ out of $2 (3 \sqrt{2}) + 2 (- \sqrt{6})$ .

$\frac{2 (4 \sqrt{3})}{2 (3 \sqrt{2} - \sqrt{6})}$

Step 12.10.2.4

Cancel the common factor.

$\frac{2 (4 \sqrt{3})}{2 (3 \sqrt{2} - \sqrt{6})}$

Step 12.10.2.5

Rewrite the expression.

$\frac{4 \sqrt{3}}{3 \sqrt{2} - \sqrt{6}}$

Step 12.11

Multiply $\frac{4 \sqrt{3}}{3 \sqrt{2} - \sqrt{6}}$ by $\frac{3 \sqrt{2} + \sqrt{6}}{3 \sqrt{2} + \sqrt{6}}$ .

$\frac{4 \sqrt{3}}{3 \sqrt{2} - \sqrt{6}} \cdot \frac{3 \sqrt{2} + \sqrt{6}}{3 \sqrt{2} + \sqrt{6}}$

Step 12.12

Multiply $\frac{4 \sqrt{3}}{3 \sqrt{2} - \sqrt{6}}$ by $\frac{3 \sqrt{2} + \sqrt{6}}{3 \sqrt{2} + \sqrt{6}}$ .

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

Step 12.13

Expand the denominator using the FOIL method.

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

Step 12.14

Simplify.

$\frac{4 \sqrt{3} (3 \sqrt{2} + \sqrt{6})}{12}$

Step 12.15

Cancel the common factor of $4$ and $12$ .

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Step 12.15.1

Factor $4$ out of $4 \sqrt{3} (3 \sqrt{2} + \sqrt{6})$ .

$\frac{4 (\sqrt{3} (3 \sqrt{2} + \sqrt{6}))}{12}$

Step 12.15.2

Cancel the common factors.

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Step 12.15.2.1

Factor $4$ out of $12$ .

$\frac{4 (\sqrt{3} (3 \sqrt{2} + \sqrt{6}))}{4 \cdot 3}$

Step 12.15.2.2

Cancel the common factor.

$\frac{4 (\sqrt{3} (3 \sqrt{2} + \sqrt{6}))}{4 \cdot 3}$

Step 12.15.2.3

Rewrite the expression.

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

Step 12.16

Apply the distributive property.

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

Step 12.17

Multiply $\sqrt{3} (3 \sqrt{2})$ .

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Step 12.17.1

Combine using the product rule for radicals.

$\frac{3 \sqrt{3 \cdot 2} + \sqrt{3} \sqrt{6}}{3}$

Step 12.17.2

Multiply $3$ by $2$ .

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

Step 12.18

Combine using the product rule for radicals.

$\frac{3 \sqrt{6} + \sqrt{3 \cdot 6}}{3}$

Step 12.19

Simplify each term.

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Step 12.19.1

Multiply $3$ by $6$ .

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

Step 12.19.2

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

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Step 12.19.2.1

Factor $9$ out of $18$ .

$\frac{3 \sqrt{6} + \sqrt{9 (2)}}{3}$

Step 12.19.2.2

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

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

Step 12.19.3

Pull terms out from under the radical.

$\frac{3 \sqrt{6} + 3 \sqrt{2}}{3}$

Step 12.20

Cancel the common factor of $3 \sqrt{6} + 3 \sqrt{2}$ and $3$ .

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Step 12.20.1

Factor $3$ out of $3 \sqrt{6}$ .

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

Step 12.20.2

Factor $3$ out of $3 \sqrt{2}$ .

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

Step 12.20.3

Factor $3$ out of $3 (\sqrt{6}) + 3 (\sqrt{2})$ .

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

Step 12.20.4

Cancel the common factors.

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Step 12.20.4.1

Factor $3$ out of $3$ .

$\frac{3 (\sqrt{6} + \sqrt{2})}{3 (1)}$

Step 12.20.4.2

Cancel the common factor.

$\frac{3 (\sqrt{6} + \sqrt{2})}{3 \cdot 1}$

Step 12.20.4.3

Rewrite the expression.

$\frac{\sqrt{6} + \sqrt{2}}{1}$

Step 12.20.4.4

Divide $\sqrt{6} + \sqrt{2}$ by $1$ .

$\sqrt{6} + \sqrt{2}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\sqrt{6} + \sqrt{2}$

Decimal Form:

$3.86370330 \dots$