Trigonometry Examples

$\sqrt{\frac{2 (40)}{32 (\frac{\sqrt{3}}{2}) (\frac{1}{2})}}$

Step 1

Reduce the expression by cancelling the common factors.

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Step 1.1

Reduce the expression $\frac{2 (40)}{32 (\frac{\sqrt{3}}{2}) \cdot (\frac{1}{2})}$ by cancelling the common factors.

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Step 1.1.1

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

$\sqrt{\frac{2 (40)}{2 (16 \frac{\sqrt{3}}{2} \cdot \frac{1}{2})}}$

Step 1.1.2

Cancel the common factor.

$\sqrt{\frac{2 \cdot 40}{2 (16 \frac{\sqrt{3}}{2} \cdot \frac{1}{2})}}$

Step 1.1.3

Rewrite the expression.

$\sqrt{\frac{40}{16 \frac{\sqrt{3}}{2} \cdot \frac{1}{2}}}$

Step 1.2

Cancel the common factor of $40$ and $16$ .

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Step 1.2.1

Factor $8$ out of $40$ .

$\sqrt{\frac{8 (5)}{16 \frac{\sqrt{3}}{2} \cdot \frac{1}{2}}}$

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

Factor $8$ out of $16 \frac{\sqrt{3}}{2} \cdot \frac{1}{2}$ .

$\sqrt{\frac{8 (5)}{8 (2 \frac{\sqrt{3}}{2} \cdot \frac{1}{2})}}$

Step 1.2.2.2

Cancel the common factor.

$\sqrt{\frac{8 \cdot 5}{8 (2 \frac{\sqrt{3}}{2} \cdot \frac{1}{2})}}$

Step 1.2.2.3

Rewrite the expression.

$\sqrt{\frac{5}{2 \frac{\sqrt{3}}{2} \cdot \frac{1}{2}}}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\sqrt{\frac{5}{\frac{2 \sqrt{3}}{2} \cdot \frac{1}{2}}}$

Step 2.2

Multiply $\frac{2 \sqrt{3}}{2}$ by $\frac{1}{2}$ .

$\sqrt{\frac{5}{\frac{2 \sqrt{3}}{2 \cdot 2}}}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Multiply $2$ by $2$ .

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

Step 3.2

Reduce the expression $\frac{2 \sqrt{3}}{4}$ by cancelling the common factors.

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Step 3.2.1

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

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

Step 3.2.2

Factor $2$ out of $4$ .

$\sqrt{\frac{5}{\frac{2 \sqrt{3}}{2 \cdot 2}}}$

Step 3.2.3

Cancel the common factor.

$\sqrt{\frac{5}{\frac{2 \sqrt{3}}{2 \cdot 2}}}$

Step 3.2.4

Rewrite the expression.

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

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

$\sqrt{5 (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})}$

Step 6

Combine and simplify the denominator.

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Step 6.1

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

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

Step 6.2

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

$\sqrt{5 \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}}$

Step 6.3

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

$\sqrt{5 \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}}$

Step 6.4

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

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

Step 6.5

Add $1$ and $1$ .

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

Step 6.6

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

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Step 6.6.1

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

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

Step 6.6.2

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

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

Step 6.6.3

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

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

Step 6.6.4

Cancel the common factor of $2$ .

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Step 6.6.4.1

Cancel the common factor.

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

Step 6.6.4.2

Rewrite the expression.

$\sqrt{5 \frac{2 \sqrt{3}}{3^{1}}}$

Step 6.6.5

Evaluate the exponent.

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

Step 7

Multiply $5 \frac{2 \sqrt{3}}{3}$ .

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Step 7.1

Combine $5$ and $\frac{2 \sqrt{3}}{3}$ .

$\sqrt{\frac{5 (2 \sqrt{3})}{3}}$

Step 7.2

Multiply $2$ by $5$ .

$\sqrt{\frac{10 \sqrt{3}}{3}}$

Step 8

Rewrite $\sqrt{\frac{10 \sqrt{3}}{3}}$ as $\frac{\sqrt{10 \sqrt{3}}}{\sqrt{3}}$ .

$\frac{\sqrt{10 \sqrt{3}}}{\sqrt{3}}$

Step 9

Multiply $\frac{\sqrt{10 \sqrt{3}}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\sqrt{10 \sqrt{3}}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 10

Combine and simplify the denominator.

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Step 10.1

Multiply $\frac{\sqrt{10 \sqrt{3}}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\sqrt{10 \sqrt{3}} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 10.2

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

$\frac{\sqrt{10 \sqrt{3}} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 10.3

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

$\frac{\sqrt{10 \sqrt{3}} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 10.4

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

$\frac{\sqrt{10 \sqrt{3}} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 10.5

Add $1$ and $1$ .

$\frac{\sqrt{10 \sqrt{3}} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 10.6

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

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Step 10.6.1

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

$\frac{\sqrt{10 \sqrt{3}} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 10.6.2

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

$\frac{\sqrt{10 \sqrt{3}} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 10.6.3

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

$\frac{\sqrt{10 \sqrt{3}} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 10.6.4

Cancel the common factor of $2$ .

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Step 10.6.4.1

Cancel the common factor.

$\frac{\sqrt{10 \sqrt{3}} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 10.6.4.2

Rewrite the expression.

$\frac{\sqrt{10 \sqrt{3}} \sqrt{3}}{3^{1}}$

Step 10.6.5

Evaluate the exponent.

$\frac{\sqrt{10 \sqrt{3}} \sqrt{3}}{3}$

Step 11

Simplify the numerator.

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Step 11.1

Combine using the product rule for radicals.

$\frac{\sqrt{10 \sqrt{3} \cdot 3}}{3}$

Step 11.2

Multiply $3$ by $10$ .

$\frac{\sqrt{30 \sqrt{3}}}{3}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{30 \sqrt{3}}}{3}$

Decimal Form:

$2.40281141 \dots$