Trigonometry Examples

$\sqrt{\frac{1}{2} \cdot (1 - \frac{10}{13})}$

Step 1

Simplify the expression.

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Step 1.1

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

$\sqrt{\frac{1}{2} (\frac{13}{13} - \frac{10}{13})}$

Step 1.2

Combine the numerators over the common denominator.

$\sqrt{\frac{1}{2} \cdot \frac{13 - 10}{13}}$

Step 1.3

Subtract $10$ from $13$ .

$\sqrt{\frac{1}{2} \cdot \frac{3}{13}}$

Step 2

Multiply $\frac{1}{2}$ by $\frac{3}{13}$ .

$\sqrt{\frac{3}{2 \cdot 13}}$

Step 3

Multiply $2$ by $13$ .

$\sqrt{\frac{3}{26}}$

Step 4

Rewrite $\sqrt{\frac{3}{26}}$ as $\frac{\sqrt{3}}{\sqrt{26}}$ .

$\frac{\sqrt{3}}{\sqrt{26}}$

Step 5

Multiply $\frac{\sqrt{3}}{\sqrt{26}}$ by $\frac{\sqrt{26}}{\sqrt{26}}$ .

$\frac{\sqrt{3}}{\sqrt{26}} \cdot \frac{\sqrt{26}}{\sqrt{26}}$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{\sqrt{3}}{\sqrt{26}}$ by $\frac{\sqrt{26}}{\sqrt{26}}$ .

$\frac{\sqrt{3} \sqrt{26}}{\sqrt{26} \sqrt{26}}$

Step 6.2

Raise $\sqrt{26}$ to the power of $1$ .

$\frac{\sqrt{3} \sqrt{26}}{{\sqrt{26}}^{1} \sqrt{26}}$

Step 6.3

Raise $\sqrt{26}$ to the power of $1$ .

$\frac{\sqrt{3} \sqrt{26}}{{\sqrt{26}}^{1} {\sqrt{26}}^{1}}$

Step 6.4

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

$\frac{\sqrt{3} \sqrt{26}}{{\sqrt{26}}^{1 + 1}}$

Step 6.5

Add $1$ and $1$ .

$\frac{\sqrt{3} \sqrt{26}}{{\sqrt{26}}^{2}}$

Step 6.6

Rewrite ${\sqrt{26}}^{2}$ as $26$ .

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Step 6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{26}$ as $26^{\frac{1}{2}}$ .

$\frac{\sqrt{3} \sqrt{26}}{{(26^{\frac{1}{2}})}^{2}}$

Step 6.6.2

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

$\frac{\sqrt{3} \sqrt{26}}{26^{\frac{1}{2} \cdot 2}}$

Step 6.6.3

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

$\frac{\sqrt{3} \sqrt{26}}{26^{\frac{2}{2}}}$

Step 6.6.4

Cancel the common factor of $2$ .

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Step 6.6.4.1

Cancel the common factor.

$\frac{\sqrt{3} \sqrt{26}}{26^{\frac{2}{2}}}$

Step 6.6.4.2

Rewrite the expression.

$\frac{\sqrt{3} \sqrt{26}}{26^{1}}$

Step 6.6.5

Evaluate the exponent.

$\frac{\sqrt{3} \sqrt{26}}{26}$

Step 7

Simplify the numerator.

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Step 7.1

Combine using the product rule for radicals.

$\frac{\sqrt{3 \cdot 26}}{26}$

Step 7.2

Multiply $3$ by $26$ .

$\frac{\sqrt{78}}{26}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{78}}{26}$

Decimal Form:

$0.33968311 \dots$