Trigonometry Examples

$\sqrt{\frac{78 - \sqrt{7}}{1} + \frac{2}{3 - \sqrt{7}}}$

Step 1

To write $\frac{78 - \sqrt{7}}{1}$ as a fraction with a common denominator, multiply by $\frac{3 - \sqrt{7}}{3 - \sqrt{7}}$ .

$\sqrt{\frac{78 - \sqrt{7}}{1} \cdot \frac{3 - \sqrt{7}}{3 - \sqrt{7}} + \frac{2}{3 - \sqrt{7}}}$

Step 2

Multiply $\frac{78 - \sqrt{7}}{1}$ by $\frac{3 - \sqrt{7}}{3 - \sqrt{7}}$ .

$\sqrt{\frac{(78 - \sqrt{7}) (3 - \sqrt{7})}{3 - \sqrt{7}} + \frac{2}{3 - \sqrt{7}}}$

Step 3

Combine the numerators over the common denominator.

$\sqrt{\frac{(78 - \sqrt{7}) (3 - \sqrt{7}) + 2}{3 - \sqrt{7}}}$

Step 4

Rewrite $\frac{(78 - \sqrt{7}) (3 - \sqrt{7}) + 2}{3 - \sqrt{7}}$ in a factored form.

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Step 4.1

Expand $(78 - \sqrt{7}) (3 - \sqrt{7})$ using the FOIL Method.

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Step 4.1.1

Apply the distributive property.

$\sqrt{\frac{78 (3 - \sqrt{7}) - \sqrt{7} (3 - \sqrt{7}) + 2}{3 - \sqrt{7}}}$

Step 4.1.2

Apply the distributive property.

$\sqrt{\frac{78 \cdot 3 + 78 (- \sqrt{7}) - \sqrt{7} (3 - \sqrt{7}) + 2}{3 - \sqrt{7}}}$

Step 4.1.3

Apply the distributive property.

$\sqrt{\frac{78 \cdot 3 + 78 (- \sqrt{7}) - \sqrt{7} \cdot 3 - \sqrt{7} (- \sqrt{7}) + 2}{3 - \sqrt{7}}}$

Step 4.2

Simplify and combine like terms.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $78$ by $3$ .

$\sqrt{\frac{234 + 78 (- \sqrt{7}) - \sqrt{7} \cdot 3 - \sqrt{7} (- \sqrt{7}) + 2}{3 - \sqrt{7}}}$

Step 4.2.1.2

Multiply $- 1$ by $78$ .

$\sqrt{\frac{234 - 78 \sqrt{7} - \sqrt{7} \cdot 3 - \sqrt{7} (- \sqrt{7}) + 2}{3 - \sqrt{7}}}$

Step 4.2.1.3

Multiply $3$ by $- 1$ .

$\sqrt{\frac{234 - 78 \sqrt{7} - 3 \sqrt{7} - \sqrt{7} (- \sqrt{7}) + 2}{3 - \sqrt{7}}}$

Step 4.2.1.4

Multiply $- \sqrt{7} (- \sqrt{7})$ .

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Step 4.2.1.4.1

Multiply $- 1$ by $- 1$ .

$\sqrt{\frac{234 - 78 \sqrt{7} - 3 \sqrt{7} + 1 \sqrt{7} \sqrt{7} + 2}{3 - \sqrt{7}}}$

Step 4.2.1.4.2

Multiply $\sqrt{7}$ by $1$ .

$\sqrt{\frac{234 - 78 \sqrt{7} - 3 \sqrt{7} + \sqrt{7} \sqrt{7} + 2}{3 - \sqrt{7}}}$

Step 4.2.1.4.3

Raise $\sqrt{7}$ to the power of $1$ .

$\sqrt{\frac{234 - 78 \sqrt{7} - 3 \sqrt{7} + {\sqrt{7}}^{1} \sqrt{7} + 2}{3 - \sqrt{7}}}$

Step 4.2.1.4.4

Raise $\sqrt{7}$ to the power of $1$ .

$\sqrt{\frac{234 - 78 \sqrt{7} - 3 \sqrt{7} + {\sqrt{7}}^{1} {\sqrt{7}}^{1} + 2}{3 - \sqrt{7}}}$

Step 4.2.1.4.5

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

$\sqrt{\frac{234 - 78 \sqrt{7} - 3 \sqrt{7} + {\sqrt{7}}^{1 + 1} + 2}{3 - \sqrt{7}}}$

Step 4.2.1.4.6

Add $1$ and $1$ .

$\sqrt{\frac{234 - 78 \sqrt{7} - 3 \sqrt{7} + {\sqrt{7}}^{2} + 2}{3 - \sqrt{7}}}$

Step 4.2.1.5

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 4.2.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\sqrt{\frac{234 - 78 \sqrt{7} - 3 \sqrt{7} + {(7^{\frac{1}{2}})}^{2} + 2}{3 - \sqrt{7}}}$

Step 4.2.1.5.2

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

$\sqrt{\frac{234 - 78 \sqrt{7} - 3 \sqrt{7} + 7^{\frac{1}{2} \cdot 2} + 2}{3 - \sqrt{7}}}$

Step 4.2.1.5.3

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

$\sqrt{\frac{234 - 78 \sqrt{7} - 3 \sqrt{7} + 7^{\frac{2}{2}} + 2}{3 - \sqrt{7}}}$

Step 4.2.1.5.4

Cancel the common factor of $2$ .

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Step 4.2.1.5.4.1

Cancel the common factor.

$\sqrt{\frac{234 - 78 \sqrt{7} - 3 \sqrt{7} + 7^{\frac{2}{2}} + 2}{3 - \sqrt{7}}}$

Step 4.2.1.5.4.2

Rewrite the expression.

$\sqrt{\frac{234 - 78 \sqrt{7} - 3 \sqrt{7} + 7^{1} + 2}{3 - \sqrt{7}}}$

Step 4.2.1.5.5

Evaluate the exponent.

$\sqrt{\frac{234 - 78 \sqrt{7} - 3 \sqrt{7} + 7 + 2}{3 - \sqrt{7}}}$

Step 4.2.2

Add $234$ and $7$ .

$\sqrt{\frac{241 - 78 \sqrt{7} - 3 \sqrt{7} + 2}{3 - \sqrt{7}}}$

Step 4.2.3

Subtract $3 \sqrt{7}$ from $- 78 \sqrt{7}$ .

$\sqrt{\frac{241 - 81 \sqrt{7} + 2}{3 - \sqrt{7}}}$

Step 4.3

Add $241$ and $2$ .

$\sqrt{\frac{243 - 81 \sqrt{7}}{3 - \sqrt{7}}}$

Step 5

Multiply $\frac{243 - 81 \sqrt{7}}{3 - \sqrt{7}}$ by $\frac{3 + \sqrt{7}}{3 + \sqrt{7}}$ .

$\sqrt{\frac{243 - 81 \sqrt{7}}{3 - \sqrt{7}} \cdot \frac{3 + \sqrt{7}}{3 + \sqrt{7}}}$

Step 6

Combine fractions.

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Step 6.1

Multiply $\frac{243 - 81 \sqrt{7}}{3 - \sqrt{7}}$ by $\frac{3 + \sqrt{7}}{3 + \sqrt{7}}$ .

$\sqrt{\frac{(243 - 81 \sqrt{7}) (3 + \sqrt{7})}{(3 - \sqrt{7}) (3 + \sqrt{7})}}$

Step 6.2

Expand the denominator using the FOIL method.

$\sqrt{\frac{(243 - 81 \sqrt{7}) (3 + \sqrt{7})}{9 + 3 \sqrt{7} - 3 \sqrt{7} - {\sqrt{7}}^{2}}}$

Step 6.3

Simplify.

$\sqrt{\frac{(243 - 81 \sqrt{7}) (3 + \sqrt{7})}{2}}$

Step 7

Simplify the numerator.

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Step 7.1

Expand $(243 - 81 \sqrt{7}) (3 + \sqrt{7})$ using the FOIL Method.

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Step 7.1.1

Apply the distributive property.

$\sqrt{\frac{243 (3 + \sqrt{7}) - 81 \sqrt{7} (3 + \sqrt{7})}{2}}$

Step 7.1.2

Apply the distributive property.

$\sqrt{\frac{243 \cdot 3 + 243 \sqrt{7} - 81 \sqrt{7} (3 + \sqrt{7})}{2}}$

Step 7.1.3

Apply the distributive property.

$\sqrt{\frac{243 \cdot 3 + 243 \sqrt{7} - 81 \sqrt{7} \cdot 3 - 81 \sqrt{7} \sqrt{7}}{2}}$

Step 7.2

Simplify and combine like terms.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Multiply $243$ by $3$ .

$\sqrt{\frac{729 + 243 \sqrt{7} - 81 \sqrt{7} \cdot 3 - 81 \sqrt{7} \sqrt{7}}{2}}$

Step 7.2.1.2

Multiply $3$ by $- 81$ .

$\sqrt{\frac{729 + 243 \sqrt{7} - 243 \sqrt{7} - 81 \sqrt{7} \sqrt{7}}{2}}$

Step 7.2.1.3

Multiply $- 81 \sqrt{7} \sqrt{7}$ .

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Step 7.2.1.3.1

Raise $\sqrt{7}$ to the power of $1$ .

$\sqrt{\frac{729 + 243 \sqrt{7} - 243 \sqrt{7} - 81 ({\sqrt{7}}^{1} \sqrt{7})}{2}}$

Step 7.2.1.3.2

Raise $\sqrt{7}$ to the power of $1$ .

$\sqrt{\frac{729 + 243 \sqrt{7} - 243 \sqrt{7} - 81 ({\sqrt{7}}^{1} {\sqrt{7}}^{1})}{2}}$

Step 7.2.1.3.3

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

$\sqrt{\frac{729 + 243 \sqrt{7} - 243 \sqrt{7} - 81 {\sqrt{7}}^{1 + 1}}{2}}$

Step 7.2.1.3.4

Add $1$ and $1$ .

$\sqrt{\frac{729 + 243 \sqrt{7} - 243 \sqrt{7} - 81 {\sqrt{7}}^{2}}{2}}$

Step 7.2.1.4

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 7.2.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\sqrt{\frac{729 + 243 \sqrt{7} - 243 \sqrt{7} - 81 {(7^{\frac{1}{2}})}^{2}}{2}}$

Step 7.2.1.4.2

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

$\sqrt{\frac{729 + 243 \sqrt{7} - 243 \sqrt{7} - 81 \cdot 7^{\frac{1}{2} \cdot 2}}{2}}$

Step 7.2.1.4.3

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

$\sqrt{\frac{729 + 243 \sqrt{7} - 243 \sqrt{7} - 81 \cdot 7^{\frac{2}{2}}}{2}}$

Step 7.2.1.4.4

Cancel the common factor of $2$ .

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Step 7.2.1.4.4.1

Cancel the common factor.

$\sqrt{\frac{729 + 243 \sqrt{7} - 243 \sqrt{7} - 81 \cdot 7^{\frac{2}{2}}}{2}}$

Step 7.2.1.4.4.2

Rewrite the expression.

$\sqrt{\frac{729 + 243 \sqrt{7} - 243 \sqrt{7} - 81 \cdot 7^{1}}{2}}$

Step 7.2.1.4.5

Evaluate the exponent.

$\sqrt{\frac{729 + 243 \sqrt{7} - 243 \sqrt{7} - 81 \cdot 7}{2}}$

Step 7.2.1.5

Multiply $- 81$ by $7$ .

$\sqrt{\frac{729 + 243 \sqrt{7} - 243 \sqrt{7} - 567}{2}}$

Step 7.2.2

Subtract $567$ from $729$ .

$\sqrt{\frac{162 + 243 \sqrt{7} - 243 \sqrt{7}}{2}}$

Step 7.2.3

Subtract $243 \sqrt{7}$ from $243 \sqrt{7}$ .

$\sqrt{\frac{162 + 0}{2}}$

Step 7.2.4

Add $162$ and $0$ .

$\sqrt{\frac{162}{2}}$

Step 8

Simplify the expression.

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Step 8.1

Divide $162$ by $2$ .

$\sqrt{81}$

Step 8.2

Rewrite $81$ as $9^{2}$ .

$\sqrt{9^{2}}$

Step 8.3

Pull terms out from under the radical, assuming positive real numbers.

$9$