Trigonometry Examples

$\sqrt{(\frac{5}{2^{2}}) + {(\frac{5}{2})}^{2}}$

Step 1

Apply the product rule to $\frac{5}{2}$ .

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

Step 2

Raise $5$ to the power of $2$ .

$\sqrt{\frac{5}{2^{2}} + \frac{25}{2^{2}}}$

Step 3

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

$\sqrt{\frac{5}{2^{2}} + \frac{25}{4}}$

Step 4

To write $\frac{5}{2^{2}}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\sqrt{\frac{5}{2^{2}} \cdot \frac{4}{4} + \frac{25}{4}}$

Step 5

To write $\frac{25}{4}$ as a fraction with a common denominator, multiply by $\frac{2^{2}}{2^{2}}$ .

$\sqrt{\frac{5}{2^{2}} \cdot \frac{4}{4} + \frac{25}{4} \cdot \frac{2^{2}}{2^{2}}}$

Step 6

Write each expression with a common denominator of $4 \cdot 2^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.1

Multiply $\frac{5}{2^{2}}$ by $\frac{4}{4}$ .

$\sqrt{\frac{5 \cdot 4}{2^{2} \cdot 4} + \frac{25}{4} \cdot \frac{2^{2}}{2^{2}}}$

Step 6.2

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

$\sqrt{\frac{5 \cdot 4}{2^{2} \cdot 2^{2}} + \frac{25}{4} \cdot \frac{2^{2}}{2^{2}}}$

Step 6.3

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

$\sqrt{\frac{5 \cdot 4}{2^{2 + 2}} + \frac{25}{4} \cdot \frac{2^{2}}{2^{2}}}$

Step 6.4

Add $2$ and $2$ .

$\sqrt{\frac{5 \cdot 4}{2^{4}} + \frac{25}{4} \cdot \frac{2^{2}}{2^{2}}}$

Step 6.5

Multiply $\frac{25}{4}$ by $\frac{2^{2}}{2^{2}}$ .

$\sqrt{\frac{5 \cdot 4}{2^{4}} + \frac{25 \cdot 2^{2}}{4 \cdot 2^{2}}}$

Step 6.6

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

$\sqrt{\frac{5 \cdot 4}{2^{4}} + \frac{25 \cdot 2^{2}}{2^{2} \cdot 2^{2}}}$

Step 6.7

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

$\sqrt{\frac{5 \cdot 4}{2^{4}} + \frac{25 \cdot 2^{2}}{2^{2 + 2}}}$

Step 6.8

Add $2$ and $2$ .

$\sqrt{\frac{5 \cdot 4}{2^{4}} + \frac{25 \cdot 2^{2}}{2^{4}}}$

Step 7

Combine the numerators over the common denominator.

$\sqrt{\frac{5 \cdot 4 + 25 \cdot 2^{2}}{2^{4}}}$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $5$ by $4$ .

$\sqrt{\frac{20 + 25 \cdot 2^{2}}{2^{4}}}$

Step 8.2

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

$\sqrt{\frac{20 + 25 \cdot 4}{2^{4}}}$

Step 8.3

Multiply $25$ by $4$ .

$\sqrt{\frac{20 + 100}{2^{4}}}$

Step 8.4

Add $20$ and $100$ .

$\sqrt{\frac{120}{2^{4}}}$

Step 9

Raise $2$ to the power of $4$ .

$\sqrt{\frac{120}{16}}$

Step 10

Cancel the common factor of $120$ and $16$ .

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Step 10.1

Factor $8$ out of $120$ .

$\sqrt{\frac{8 (15)}{16}}$

Step 10.2

Cancel the common factors.

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Step 10.2.1

Factor $8$ out of $16$ .

$\sqrt{\frac{8 \cdot 15}{8 \cdot 2}}$

Step 10.2.2

Cancel the common factor.

$\sqrt{\frac{8 \cdot 15}{8 \cdot 2}}$

Step 10.2.3

Rewrite the expression.

$\sqrt{\frac{15}{2}}$

Step 11

Rewrite $\sqrt{\frac{15}{2}}$ as $\frac{\sqrt{15}}{\sqrt{2}}$ .

$\frac{\sqrt{15}}{\sqrt{2}}$

Step 12

Multiply $\frac{\sqrt{15}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{15}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 13

Combine and simplify the denominator.

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Step 13.1

Multiply $\frac{\sqrt{15}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{15} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 13.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{15} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 13.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{15} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 13.4

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

$\frac{\sqrt{15} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 13.5

Add $1$ and $1$ .

$\frac{\sqrt{15} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 13.6

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

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Step 13.6.1

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

$\frac{\sqrt{15} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 13.6.2

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

$\frac{\sqrt{15} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 13.6.3

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

$\frac{\sqrt{15} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 13.6.4

Cancel the common factor of $2$ .

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Step 13.6.4.1

Cancel the common factor.

$\frac{\sqrt{15} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 13.6.4.2

Rewrite the expression.

$\frac{\sqrt{15} \sqrt{2}}{2^{1}}$

Step 13.6.5

Evaluate the exponent.

$\frac{\sqrt{15} \sqrt{2}}{2}$

Step 14

Simplify the numerator.

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Step 14.1

Combine using the product rule for radicals.

$\frac{\sqrt{15 \cdot 2}}{2}$

Step 14.2

Multiply $15$ by $2$ .

$\frac{\sqrt{30}}{2}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{30}}{2}$

Decimal Form:

$2.73861278 \dots$