Trigonometry Examples

$\sqrt{\frac{1}{2} \cdot (1 - \frac{\sqrt{48}}{7})}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 1.1.1

Factor $16$ out of $48$ .

$\sqrt{\frac{1}{2} (1 - \frac{\sqrt{16 (3)}}{7})}$

Step 1.1.2

Rewrite $16$ as $4^{2}$ .

$\sqrt{\frac{1}{2} (1 - \frac{\sqrt{4^{2} \cdot 3}}{7})}$

Step 1.2

Pull terms out from under the radical.

$\sqrt{\frac{1}{2} (1 - \frac{4 \sqrt{3}}{7})}$

Step 2

Combine fractions.

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Step 2.1

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

$\sqrt{\frac{1}{2} (\frac{7}{7} - \frac{4 \sqrt{3}}{7})}$

Step 2.2

Combine the numerators over the common denominator.

$\sqrt{\frac{1}{2} \cdot \frac{7 - 4 \sqrt{3}}{7}}$

Step 2.3

Multiply $\frac{1}{2}$ by $\frac{7 - 4 \sqrt{3}}{7}$ .

$\sqrt{\frac{7 - 4 \sqrt{3}}{2 \cdot 7}}$

Step 2.4

Multiply $2$ by $7$ .

$\sqrt{\frac{7 - 4 \sqrt{3}}{14}}$

Step 3

Rewrite $\sqrt{\frac{7 - 4 \sqrt{3}}{14}}$ as $\frac{\sqrt{7 - 4 \sqrt{3}}}{\sqrt{14}}$ .

$\frac{\sqrt{7 - 4 \sqrt{3}}}{\sqrt{14}}$

Step 4

Multiply $\frac{\sqrt{7 - 4 \sqrt{3}}}{\sqrt{14}}$ by $\frac{\sqrt{14}}{\sqrt{14}}$ .

$\frac{\sqrt{7 - 4 \sqrt{3}}}{\sqrt{14}} \cdot \frac{\sqrt{14}}{\sqrt{14}}$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{\sqrt{7 - 4 \sqrt{3}}}{\sqrt{14}}$ by $\frac{\sqrt{14}}{\sqrt{14}}$ .

$\frac{\sqrt{7 - 4 \sqrt{3}} \sqrt{14}}{\sqrt{14} \sqrt{14}}$

Step 5.2

Raise $\sqrt{14}$ to the power of $1$ .

$\frac{\sqrt{7 - 4 \sqrt{3}} \sqrt{14}}{{\sqrt{14}}^{1} \sqrt{14}}$

Step 5.3

Raise $\sqrt{14}$ to the power of $1$ .

$\frac{\sqrt{7 - 4 \sqrt{3}} \sqrt{14}}{{\sqrt{14}}^{1} {\sqrt{14}}^{1}}$

Step 5.4

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

$\frac{\sqrt{7 - 4 \sqrt{3}} \sqrt{14}}{{\sqrt{14}}^{1 + 1}}$

Step 5.5

Add $1$ and $1$ .

$\frac{\sqrt{7 - 4 \sqrt{3}} \sqrt{14}}{{\sqrt{14}}^{2}}$

Step 5.6

Rewrite ${\sqrt{14}}^{2}$ as $14$ .

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Step 5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{14}$ as $14^{\frac{1}{2}}$ .

$\frac{\sqrt{7 - 4 \sqrt{3}} \sqrt{14}}{{(14^{\frac{1}{2}})}^{2}}$

Step 5.6.2

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

$\frac{\sqrt{7 - 4 \sqrt{3}} \sqrt{14}}{14^{\frac{1}{2} \cdot 2}}$

Step 5.6.3

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

$\frac{\sqrt{7 - 4 \sqrt{3}} \sqrt{14}}{14^{\frac{2}{2}}}$

Step 5.6.4

Cancel the common factor of $2$ .

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Step 5.6.4.1

Cancel the common factor.

$\frac{\sqrt{7 - 4 \sqrt{3}} \sqrt{14}}{14^{\frac{2}{2}}}$

Step 5.6.4.2

Rewrite the expression.

$\frac{\sqrt{7 - 4 \sqrt{3}} \sqrt{14}}{14^{1}}$

Step 5.6.5

Evaluate the exponent.

$\frac{\sqrt{7 - 4 \sqrt{3}} \sqrt{14}}{14}$

Step 6

Combine using the product rule for radicals.

$\frac{\sqrt{(7 - 4 \sqrt{3}) \cdot 14}}{14}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{(7 - 4 \sqrt{3}) \cdot 14}}{14}$

Decimal Form:

$0.07161243 \dots$