Trigonometry Examples

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

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

Step 1

Apply the product rule to

\frac{5}{2}

$\frac{5}{2}$ .

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

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

Step 2

Raise

5

$5$ to the power of

2

$2$ .

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

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

Step 3

Raise

2

$2$ to the power of

2

$2$ .

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

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

Step 4

To write

\frac{5}{2^{2}}

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

\frac{4}{4}

$\frac{4}{4}$ .

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

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

Step 5

To write

\frac{25}{4}

$\frac{25}{4}$ as a fraction with a common denominator, multiply by

\frac{2^{2}}{2^{2}}

$\frac{2^{2}}{2^{2}}$ .

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

$4 \cdot 2^{2}$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 6.1

Multiply

\frac{5}{2^{2}}

$\frac{5}{2^{2}}$ by

\frac{4}{4}

$\frac{4}{4}$ .

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

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

Step 6.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

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

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 6.4

Add

2

$2$ and

2

$2$ .

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

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

Step 6.5

Multiply

\frac{25}{4}

$\frac{25}{4}$ by

\frac{2^{2}}{2^{2}}

$\frac{2^{2}}{2^{2}}$ .

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

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

Step 6.6

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

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

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 6.8

Add

2

$2$ and

2

$2$ .

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

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

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

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

Step 7

Combine the numerators over the common denominator.

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

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

Step 8

Simplify the numerator.

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Step 8.1

Multiply

5

$5$ by

4

$4$ .

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

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

Step 8.2

Raise

2

$2$ to the power of

2

$2$ .

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

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

Step 8.3

Multiply

25

$25$ by

4

$4$ .

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

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

Step 8.4

Add

20

$20$ and

100

$100$ .

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

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

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

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

Step 9

Raise

2

$2$ to the power of

4

$4$ .

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

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

Step 10

Cancel the common factor of

120

$120$ and

16

$16$ .

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Step 10.1

Factor

8

$8$ out of

120

$120$ .

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

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

Step 10.2

Cancel the common factors.

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Step 10.2.1

Factor

8

$8$ out of

16

$16$ .

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

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

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:

$2.73861278 \dots$