Trigonometry Examples

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

Step 1

Combine $\frac{5}{3}$ and $\sqrt{\frac{1}{3}}$ .

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

Step 2

Simplify the numerator.

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Step 2.1

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

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

Step 2.2

Any root of $1$ is $1$ .

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

Step 2.3

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

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

Step 2.4

Combine and simplify the denominator.

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.4.4

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

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

Step 2.4.5

Add $1$ and $1$ .

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

Step 2.4.6

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

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Step 2.4.6.1

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

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

Step 2.4.6.2

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

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

Step 2.4.6.3

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

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

Step 2.4.6.4

Cancel the common factor of $2$ .

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Step 2.4.6.4.1

Cancel the common factor.

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

Step 2.4.6.4.2

Rewrite the expression.

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

Step 2.4.6.5

Evaluate the exponent.

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

Step 3

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

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Multiply $\frac{5 \sqrt{3}}{3} \cdot \frac{1}{3}$ .

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Step 5.1

Multiply $\frac{5 \sqrt{3}}{3}$ by $\frac{1}{3}$ .

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

Step 5.2

Multiply $3$ by $3$ .

$\sqrt{\frac{5 \sqrt{3}}{9}}$

Step 6

Rewrite $\sqrt{\frac{5 \sqrt{3}}{9}}$ as $\frac{\sqrt{5 \sqrt{3}}}{\sqrt{9}}$ .

$\frac{\sqrt{5 \sqrt{3}}}{\sqrt{9}}$

Step 7

Simplify the denominator.

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Step 7.1

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

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

Step 7.2

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

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.98094365 \dots$