Trigonometry Examples

$\sqrt{\frac{5}{9} - \frac{11}{27}}$

Step 1

To write $\frac{5}{9}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\sqrt{\frac{5}{9} \cdot \frac{3}{3} - \frac{11}{27}}$

Step 2

Write each expression with a common denominator of $27$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.1

Multiply $\frac{5}{9}$ by $\frac{3}{3}$ .

$\sqrt{\frac{5 \cdot 3}{9 \cdot 3} - \frac{11}{27}}$

Step 2.2

Multiply $9$ by $3$ .

$\sqrt{\frac{5 \cdot 3}{27} - \frac{11}{27}}$

Step 3

Combine the numerators over the common denominator.

$\sqrt{\frac{5 \cdot 3 - 11}{27}}$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $5$ by $3$ .

$\sqrt{\frac{15 - 11}{27}}$

Step 4.2

Subtract $11$ from $15$ .

$\sqrt{\frac{4}{27}}$

Step 5

Rewrite $\sqrt{\frac{4}{27}}$ as $\frac{\sqrt{4}}{\sqrt{27}}$ .

$\frac{\sqrt{4}}{\sqrt{27}}$

Step 6

Simplify the numerator.

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Step 6.1

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

$\frac{\sqrt{2^{2}}}{\sqrt{27}}$

Step 6.2

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

$\frac{2}{\sqrt{27}}$

Step 7

Simplify the denominator.

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Step 7.1

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 7.1.1

Factor $9$ out of $27$ .

$\frac{2}{\sqrt{9 (3)}}$

Step 7.1.2

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

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

Step 7.2

Pull terms out from under the radical.

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

Step 8

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

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

Step 9

Combine and simplify the denominator.

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Step 9.1

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

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

Step 9.2

Move $\sqrt{3}$ .

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

Step 9.3

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

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

Step 9.4

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

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

Step 9.5

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

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

Step 9.6

Add $1$ and $1$ .

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

Step 9.7

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

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Step 9.7.1

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

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

Step 9.7.2

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

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

Step 9.7.3

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

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

Step 9.7.4

Cancel the common factor of $2$ .

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Step 9.7.4.1

Cancel the common factor.

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

Step 9.7.4.2

Rewrite the expression.

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

Step 9.7.5

Evaluate the exponent.

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

Step 10

Multiply $3$ by $3$ .

$\frac{2 \sqrt{3}}{9}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{3}}{9}$

Decimal Form:

$0.38490017 \dots$