Trigonometry Examples

$\sqrt{\frac{2 (50)}{32 (\frac{1}{2}) (\sqrt{3})}}$

Step 1

Reduce the expression by cancelling the common factors.

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Step 1.1

Reduce the expression $\frac{2 (50)}{32 (\frac{1}{2}) \cdot (\sqrt{3})}$ by cancelling the common factors.

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Step 1.1.1

Factor $2$ out of $32 (\frac{1}{2}) \cdot (\sqrt{3})$ .

$\sqrt{\frac{2 (50)}{2 (16 (\frac{1}{2}) \cdot \sqrt{3})}}$

Step 1.1.2

Cancel the common factor.

$\sqrt{\frac{2 \cdot 50}{2 (16 (\frac{1}{2}) \cdot \sqrt{3})}}$

Step 1.1.3

Rewrite the expression.

$\sqrt{\frac{50}{16 (\frac{1}{2}) \cdot \sqrt{3}}}$

Step 1.2

Cancel the common factor of $50$ and $16$ .

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Step 1.2.1

Factor $2$ out of $50$ .

$\sqrt{\frac{2 (25)}{16 (\frac{1}{2}) \cdot \sqrt{3}}}$

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

Factor $2$ out of $16 (\frac{1}{2}) \cdot \sqrt{3}$ .

$\sqrt{\frac{2 (25)}{2 (8 (\frac{1}{2}) \cdot \sqrt{3})}}$

Step 1.2.2.2

Cancel the common factor.

$\sqrt{\frac{2 \cdot 25}{2 (8 (\frac{1}{2}) \cdot \sqrt{3})}}$

Step 1.2.2.3

Rewrite the expression.

$\sqrt{\frac{25}{8 (\frac{1}{2}) \cdot \sqrt{3}}}$

Step 2

Simplify the denominator.

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Step 2.1

Combine $8$ and $\frac{1}{2}$ .

$\sqrt{\frac{25}{\frac{8}{2} \sqrt{3}}}$

Step 2.2

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

$\sqrt{\frac{25}{\frac{8 \sqrt{3}}{2}}}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Reduce the expression $\frac{8 \sqrt{3}}{2}$ by cancelling the common factors.

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Step 3.1.1

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

$\sqrt{\frac{25}{\frac{2 (4 \sqrt{3})}{2}}}$

Step 3.1.2

Factor $2$ out of $2$ .

$\sqrt{\frac{25}{\frac{2 (4 \sqrt{3})}{2 (1)}}}$

Step 3.1.3

Cancel the common factor.

$\sqrt{\frac{25}{\frac{2 (4 \sqrt{3})}{2 \cdot 1}}}$

Step 3.1.4

Rewrite the expression.

$\sqrt{\frac{25}{\frac{4 \sqrt{3}}{1}}}$

Step 3.2

Divide $4 \sqrt{3}$ by $1$ .

$\sqrt{\frac{25}{4 \sqrt{3}}}$

Step 4

Multiply $\frac{25}{4 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\sqrt{\frac{25}{4 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}}$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{25}{4 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\sqrt{\frac{25 \sqrt{3}}{4 \sqrt{3} \sqrt{3}}}$

Step 5.2

Move $\sqrt{3}$ .

$\sqrt{\frac{25 \sqrt{3}}{4 (\sqrt{3} \sqrt{3})}}$

Step 5.3

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

$\sqrt{\frac{25 \sqrt{3}}{4 ({\sqrt{3}}^{1} \sqrt{3})}}$

Step 5.4

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

$\sqrt{\frac{25 \sqrt{3}}{4 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}}$

Step 5.5

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

$\sqrt{\frac{25 \sqrt{3}}{4 {\sqrt{3}}^{1 + 1}}}$

Step 5.6

Add $1$ and $1$ .

$\sqrt{\frac{25 \sqrt{3}}{4 {\sqrt{3}}^{2}}}$

Step 5.7

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

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Step 5.7.1

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

$\sqrt{\frac{25 \sqrt{3}}{4 {(3^{\frac{1}{2}})}^{2}}}$

Step 5.7.2

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

$\sqrt{\frac{25 \sqrt{3}}{4 \cdot 3^{\frac{1}{2} \cdot 2}}}$

Step 5.7.3

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

$\sqrt{\frac{25 \sqrt{3}}{4 \cdot 3^{\frac{2}{2}}}}$

Step 5.7.4

Cancel the common factor of $2$ .

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Step 5.7.4.1

Cancel the common factor.

$\sqrt{\frac{25 \sqrt{3}}{4 \cdot 3^{\frac{2}{2}}}}$

Step 5.7.4.2

Rewrite the expression.

$\sqrt{\frac{25 \sqrt{3}}{4 \cdot 3^{1}}}$

Step 5.7.5

Evaluate the exponent.

$\sqrt{\frac{25 \sqrt{3}}{4 \cdot 3}}$

Step 6

Multiply $4$ by $3$ .

$\sqrt{\frac{25 \sqrt{3}}{12}}$

Step 7

Rewrite $\sqrt{\frac{25 \sqrt{3}}{12}}$ as $\frac{\sqrt{25 \sqrt{3}}}{\sqrt{12}}$ .

$\frac{\sqrt{25 \sqrt{3}}}{\sqrt{12}}$

Step 8

Simplify the numerator.

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Step 8.1

Rewrite $25$ as $5^{2}$ .

$\frac{\sqrt{5^{2} \sqrt{3}}}{\sqrt{12}}$

Step 8.2

Pull terms out from under the radical.

$\frac{5 \sqrt{\sqrt{3}}}{\sqrt{12}}$

Step 8.3

Rewrite $\sqrt{\sqrt{3}}$ as $\sqrt[4]{3}$ .

$\frac{5 \sqrt[4]{3}}{\sqrt{12}}$

Step 9

Simplify the denominator.

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Step 9.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 9.1.1

Factor $4$ out of $12$ .

$\frac{5 \sqrt[4]{3}}{\sqrt{4 (3)}}$

Step 9.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{5 \sqrt[4]{3}}{\sqrt{2^{2} \cdot 3}}$

Step 9.2

Pull terms out from under the radical.

$\frac{5 \sqrt[4]{3}}{2 \sqrt{3}}$

Step 10

Multiply $\frac{5 \sqrt[4]{3}}{2 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{5 \sqrt[4]{3}}{2 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 11

Combine and simplify the denominator.

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Step 11.1

Multiply $\frac{5 \sqrt[4]{3}}{2 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{5 \sqrt[4]{3} \sqrt{3}}{2 \sqrt{3} \sqrt{3}}$

Step 11.2

Move $\sqrt{3}$ .

$\frac{5 \sqrt[4]{3} \sqrt{3}}{2 (\sqrt{3} \sqrt{3})}$

Step 11.3

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

$\frac{5 \sqrt[4]{3} \sqrt{3}}{2 ({\sqrt{3}}^{1} \sqrt{3})}$

Step 11.4

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

$\frac{5 \sqrt[4]{3} \sqrt{3}}{2 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}$

Step 11.5

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

$\frac{5 \sqrt[4]{3} \sqrt{3}}{2 {\sqrt{3}}^{1 + 1}}$

Step 11.6

Add $1$ and $1$ .

$\frac{5 \sqrt[4]{3} \sqrt{3}}{2 {\sqrt{3}}^{2}}$

Step 11.7

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

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Step 11.7.1

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

$\frac{5 \sqrt[4]{3} \sqrt{3}}{2 {(3^{\frac{1}{2}})}^{2}}$

Step 11.7.2

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

$\frac{5 \sqrt[4]{3} \sqrt{3}}{2 \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 11.7.3

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

$\frac{5 \sqrt[4]{3} \sqrt{3}}{2 \cdot 3^{\frac{2}{2}}}$

Step 11.7.4

Cancel the common factor of $2$ .

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Step 11.7.4.1

Cancel the common factor.

$\frac{5 \sqrt[4]{3} \sqrt{3}}{2 \cdot 3^{\frac{2}{2}}}$

Step 11.7.4.2

Rewrite the expression.

$\frac{5 \sqrt[4]{3} \sqrt{3}}{2 \cdot 3^{1}}$

Step 11.7.5

Evaluate the exponent.

$\frac{5 \sqrt[4]{3} \sqrt{3}}{2 \cdot 3}$

Step 12

Simplify the numerator.

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Step 12.1

Rewrite the expression using the least common index of $4$ .

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Step 12.1.1

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

$\frac{5 (3^{\frac{1}{2}} \sqrt[4]{3})}{2 \cdot 3}$

Step 12.1.2

Rewrite $3^{\frac{1}{2}}$ as $3^{\frac{2}{4}}$ .

$\frac{5 (3^{\frac{2}{4}} \sqrt[4]{3})}{2 \cdot 3}$

Step 12.1.3

Rewrite $3^{\frac{2}{4}}$ as $\sqrt[4]{3^{2}}$ .

$\frac{5 (\sqrt[4]{3^{2}} \sqrt[4]{3})}{2 \cdot 3}$

Step 12.2

Combine using the product rule for radicals.

$\frac{5 \sqrt[4]{3^{2} \cdot 3}}{2 \cdot 3}$

Step 12.3

Multiply $3^{2}$ by $3$ by adding the exponents.

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Step 12.3.1

Multiply $3^{2}$ by $3$ .

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Step 12.3.1.1

Raise $3$ to the power of $1$ .

$\frac{5 \sqrt[4]{3^{2} \cdot 3^{1}}}{2 \cdot 3}$

Step 12.3.1.2

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

$\frac{5 \sqrt[4]{3^{2 + 1}}}{2 \cdot 3}$

Step 12.3.2

Add $2$ and $1$ .

$\frac{5 \sqrt[4]{3^{3}}}{2 \cdot 3}$

Step 13

Simplify the expression.

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Step 13.1

Multiply $2$ by $3$ .

$\frac{5 \sqrt[4]{3^{3}}}{6}$

Step 13.2

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

$\frac{5 \sqrt[4]{27}}{6}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{5 \sqrt[4]{27}}{6}$

Decimal Form:

$1.89958921 \dots$