Trigonometry Examples

$\sqrt{\frac{1 - \frac{8}{\sqrt{73}}}{\frac{3}{\sqrt{73}}}}$

Step 1

Multiply $\frac{8}{\sqrt{73}}$ by $\frac{\sqrt{73}}{\sqrt{73}}$ .

$\sqrt{\frac{1 - (\frac{8}{\sqrt{73}} \cdot \frac{\sqrt{73}}{\sqrt{73}})}{\frac{3}{\sqrt{73}}}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{8}{\sqrt{73}}$ by $\frac{\sqrt{73}}{\sqrt{73}}$ .

$\sqrt{\frac{1 - \frac{8 \sqrt{73}}{\sqrt{73} \sqrt{73}}}{\frac{3}{\sqrt{73}}}}$

Step 2.2

Raise $\sqrt{73}$ to the power of $1$ .

$\sqrt{\frac{1 - \frac{8 \sqrt{73}}{{\sqrt{73}}^{1} \sqrt{73}}}{\frac{3}{\sqrt{73}}}}$

Step 2.3

Raise $\sqrt{73}$ to the power of $1$ .

$\sqrt{\frac{1 - \frac{8 \sqrt{73}}{{\sqrt{73}}^{1} {\sqrt{73}}^{1}}}{\frac{3}{\sqrt{73}}}}$

Step 2.4

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

$\sqrt{\frac{1 - \frac{8 \sqrt{73}}{{\sqrt{73}}^{1 + 1}}}{\frac{3}{\sqrt{73}}}}$

Step 2.5

Add $1$ and $1$ .

$\sqrt{\frac{1 - \frac{8 \sqrt{73}}{{\sqrt{73}}^{2}}}{\frac{3}{\sqrt{73}}}}$

Step 2.6

Rewrite ${\sqrt{73}}^{2}$ as $73$ .

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Step 2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{73}$ as $73^{\frac{1}{2}}$ .

$\sqrt{\frac{1 - \frac{8 \sqrt{73}}{{(73^{\frac{1}{2}})}^{2}}}{\frac{3}{\sqrt{73}}}}$

Step 2.6.2

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

$\sqrt{\frac{1 - \frac{8 \sqrt{73}}{73^{\frac{1}{2} \cdot 2}}}{\frac{3}{\sqrt{73}}}}$

Step 2.6.3

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

$\sqrt{\frac{1 - \frac{8 \sqrt{73}}{73^{\frac{2}{2}}}}{\frac{3}{\sqrt{73}}}}$

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

$\sqrt{\frac{1 - \frac{8 \sqrt{73}}{73^{\frac{2}{2}}}}{\frac{3}{\sqrt{73}}}}$

Step 2.6.4.2

Rewrite the expression.

$\sqrt{\frac{1 - \frac{8 \sqrt{73}}{73^{1}}}{\frac{3}{\sqrt{73}}}}$

Step 2.6.5

Evaluate the exponent.

$\sqrt{\frac{1 - \frac{8 \sqrt{73}}{73}}{\frac{3}{\sqrt{73}}}}$

Step 3

Simplify the expression.

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Step 3.1

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

$\sqrt{\frac{\frac{73}{73} - \frac{8 \sqrt{73}}{73}}{\frac{3}{\sqrt{73}}}}$

Step 3.2

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{73 - 8 \sqrt{73}}{73}}{\frac{3}{\sqrt{73}}}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{73 - 8 \sqrt{73}}{73} \cdot \frac{\sqrt{73}}{3}}$

Step 5

Multiply $\frac{73 - 8 \sqrt{73}}{73} \cdot \frac{\sqrt{73}}{3}$ .

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Step 5.1

Multiply $\frac{73 - 8 \sqrt{73}}{73}$ by $\frac{\sqrt{73}}{3}$ .

$\sqrt{\frac{(73 - 8 \sqrt{73}) \sqrt{73}}{73 \cdot 3}}$

Step 5.2

Multiply $73$ by $3$ .

$\sqrt{\frac{(73 - 8 \sqrt{73}) \sqrt{73}}{219}}$

Step 6

Apply the distributive property.

$\sqrt{\frac{73 \sqrt{73} - 8 \sqrt{73} \sqrt{73}}{219}}$

Step 7

Multiply $- 8 \sqrt{73} \sqrt{73}$ .

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Step 7.1

Raise $\sqrt{73}$ to the power of $1$ .

$\sqrt{\frac{73 \sqrt{73} - 8 ({\sqrt{73}}^{1} \sqrt{73})}{219}}$

Step 7.2

Raise $\sqrt{73}$ to the power of $1$ .

$\sqrt{\frac{73 \sqrt{73} - 8 ({\sqrt{73}}^{1} {\sqrt{73}}^{1})}{219}}$

Step 7.3

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

$\sqrt{\frac{73 \sqrt{73} - 8 {\sqrt{73}}^{1 + 1}}{219}}$

Step 7.4

Add $1$ and $1$ .

$\sqrt{\frac{73 \sqrt{73} - 8 {\sqrt{73}}^{2}}{219}}$

Step 8

Simplify each term.

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Step 8.1

Rewrite ${\sqrt{73}}^{2}$ as $73$ .

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Step 8.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{73}$ as $73^{\frac{1}{2}}$ .

$\sqrt{\frac{73 \sqrt{73} - 8 {(73^{\frac{1}{2}})}^{2}}{219}}$

Step 8.1.2

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

$\sqrt{\frac{73 \sqrt{73} - 8 \cdot 73^{\frac{1}{2} \cdot 2}}{219}}$

Step 8.1.3

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

$\sqrt{\frac{73 \sqrt{73} - 8 \cdot 73^{\frac{2}{2}}}{219}}$

Step 8.1.4

Cancel the common factor of $2$ .

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Step 8.1.4.1

Cancel the common factor.

$\sqrt{\frac{73 \sqrt{73} - 8 \cdot 73^{\frac{2}{2}}}{219}}$

Step 8.1.4.2

Rewrite the expression.

$\sqrt{\frac{73 \sqrt{73} - 8 \cdot 73^{1}}{219}}$

Step 8.1.5

Evaluate the exponent.

$\sqrt{\frac{73 \sqrt{73} - 8 \cdot 73}{219}}$

Step 8.2

Multiply $- 8$ by $73$ .

$\sqrt{\frac{73 \sqrt{73} - 584}{219}}$

Step 9

Cancel the common factor of $73 \sqrt{73} - 584$ and $219$ .

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Step 9.1

Factor $73$ out of $73 \sqrt{73}$ .

$\sqrt{\frac{73 (\sqrt{73}) - 584}{219}}$

Step 9.2

Factor $73$ out of $- 584$ .

$\sqrt{\frac{73 \sqrt{73} + 73 \cdot - 8}{219}}$

Step 9.3

Factor $73$ out of $73 \sqrt{73} + 73 \cdot - 8$ .

$\sqrt{\frac{73 (\sqrt{73} - 8)}{219}}$

Step 9.4

Cancel the common factors.

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Step 9.4.1

Factor $73$ out of $219$ .

$\sqrt{\frac{73 (\sqrt{73} - 8)}{73 (3)}}$

Step 9.4.2

Cancel the common factor.

$\sqrt{\frac{73 (\sqrt{73} - 8)}{73 \cdot 3}}$

Step 9.4.3

Rewrite the expression.

$\sqrt{\frac{\sqrt{73} - 8}{3}}$

Step 10

Rewrite $\sqrt{\frac{\sqrt{73} - 8}{3}}$ as $\frac{\sqrt{\sqrt{73} - 8}}{\sqrt{3}}$ .

$\frac{\sqrt{\sqrt{73} - 8}}{\sqrt{3}}$

Step 11

Multiply $\frac{\sqrt{\sqrt{73} - 8}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\sqrt{\sqrt{73} - 8}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 12

Combine and simplify the denominator.

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Step 12.1

Multiply $\frac{\sqrt{\sqrt{73} - 8}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\sqrt{\sqrt{73} - 8} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 12.2

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

$\frac{\sqrt{\sqrt{73} - 8} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 12.3

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

$\frac{\sqrt{\sqrt{73} - 8} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 12.4

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

$\frac{\sqrt{\sqrt{73} - 8} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 12.5

Add $1$ and $1$ .

$\frac{\sqrt{\sqrt{73} - 8} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 12.6

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

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Step 12.6.1

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

$\frac{\sqrt{\sqrt{73} - 8} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 12.6.2

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

$\frac{\sqrt{\sqrt{73} - 8} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 12.6.3

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

$\frac{\sqrt{\sqrt{73} - 8} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 12.6.4

Cancel the common factor of $2$ .

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Step 12.6.4.1

Cancel the common factor.

$\frac{\sqrt{\sqrt{73} - 8} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 12.6.4.2

Rewrite the expression.

$\frac{\sqrt{\sqrt{73} - 8} \sqrt{3}}{3^{1}}$

Step 12.6.5

Evaluate the exponent.

$\frac{\sqrt{\sqrt{73} - 8} \sqrt{3}}{3}$

Step 13

Combine using the product rule for radicals.

$\frac{\sqrt{(\sqrt{73} - 8) \cdot 3}}{3}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{(\sqrt{73} - 8) \cdot 3}}{3}$

Decimal Form:

$0.42583398 \dots$