Trigonometry Examples

$\sqrt{{(\frac{\sqrt{47}}{2})}^{2} + {(\frac{\sqrt{53}}{2})}^{2}}$

Step 1

Apply the product rule to $\frac{\sqrt{47}}{2}$ .

$\sqrt{\frac{{\sqrt{47}}^{2}}{2^{2}} + {(\frac{\sqrt{53}}{2})}^{2}}$

Step 2

Rewrite ${\sqrt{47}}^{2}$ as $47$ .

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{47}$ as $47^{\frac{1}{2}}$ .

$\sqrt{\frac{{(47^{\frac{1}{2}})}^{2}}{2^{2}} + {(\frac{\sqrt{53}}{2})}^{2}}$

Step 2.2

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

$\sqrt{\frac{47^{\frac{1}{2} \cdot 2}}{2^{2}} + {(\frac{\sqrt{53}}{2})}^{2}}$

Step 2.3

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

$\sqrt{\frac{47^{\frac{2}{2}}}{2^{2}} + {(\frac{\sqrt{53}}{2})}^{2}}$

Step 2.4

Cancel the common factor of $2$ .

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Step 2.4.1

Cancel the common factor.

$\sqrt{\frac{47^{\frac{2}{2}}}{2^{2}} + {(\frac{\sqrt{53}}{2})}^{2}}$

Step 2.4.2

Rewrite the expression.

$\sqrt{\frac{47^{1}}{2^{2}} + {(\frac{\sqrt{53}}{2})}^{2}}$

Step 2.5

Evaluate the exponent.

$\sqrt{\frac{47}{2^{2}} + {(\frac{\sqrt{53}}{2})}^{2}}$

Step 3

Simplify the expression.

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Step 3.1

Raise $2$ to the power of $2$ .

$\sqrt{\frac{47}{4} + {(\frac{\sqrt{53}}{2})}^{2}}$

Step 3.2

Apply the product rule to $\frac{\sqrt{53}}{2}$ .

$\sqrt{\frac{47}{4} + \frac{{\sqrt{53}}^{2}}{2^{2}}}$

Step 4

Rewrite ${\sqrt{53}}^{2}$ as $53$ .

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{53}$ as $53^{\frac{1}{2}}$ .

$\sqrt{\frac{47}{4} + \frac{{(53^{\frac{1}{2}})}^{2}}{2^{2}}}$

Step 4.2

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

$\sqrt{\frac{47}{4} + \frac{53^{\frac{1}{2} \cdot 2}}{2^{2}}}$

Step 4.3

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

$\sqrt{\frac{47}{4} + \frac{53^{\frac{2}{2}}}{2^{2}}}$

Step 4.4

Cancel the common factor of $2$ .

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Step 4.4.1

Cancel the common factor.

$\sqrt{\frac{47}{4} + \frac{53^{\frac{2}{2}}}{2^{2}}}$

Step 4.4.2

Rewrite the expression.

$\sqrt{\frac{47}{4} + \frac{53^{1}}{2^{2}}}$

Step 4.5

Evaluate the exponent.

$\sqrt{\frac{47}{4} + \frac{53}{2^{2}}}$

Step 5

Simplify the expression.

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Step 5.1

Raise $2$ to the power of $2$ .

$\sqrt{\frac{47}{4} + \frac{53}{4}}$

Step 5.2

Combine the numerators over the common denominator.

$\sqrt{\frac{47 + 53}{4}}$

Step 5.3

Add $47$ and $53$ .

$\sqrt{\frac{100}{4}}$

Step 5.4

Divide $100$ by $4$ .

$\sqrt{25}$

Step 5.5

Rewrite $25$ as $5^{2}$ .

$\sqrt{5^{2}}$

Step 6

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

$5$