Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.2

Simplify the numerator.

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Step 1.2.1

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

$v (\frac{25 {\sqrt{2}}^{2}}{2^{2}} + {(- \frac{5 \sqrt{2}}{2})}^{2})$

Step 1.2.2

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

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Step 1.2.2.1

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

$v (\frac{25 {(2^{\frac{1}{2}})}^{2}}{2^{2}} + {(- \frac{5 \sqrt{2}}{2})}^{2})$

Step 1.2.2.2

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

$v (\frac{25 \cdot 2^{\frac{1}{2} \cdot 2}}{2^{2}} + {(- \frac{5 \sqrt{2}}{2})}^{2})$

Step 1.2.2.3

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

$v (\frac{25 \cdot 2^{\frac{2}{2}}}{2^{2}} + {(- \frac{5 \sqrt{2}}{2})}^{2})$

Step 1.2.2.4

Cancel the common factor of $2$ .

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Step 1.2.2.4.1

Cancel the common factor.

$v (\frac{25 \cdot 2^{\frac{2}{2}}}{2^{2}} + {(- \frac{5 \sqrt{2}}{2})}^{2})$

Step 1.2.2.4.2

Rewrite the expression.

$v (\frac{25 \cdot 2^{1}}{2^{2}} + {(- \frac{5 \sqrt{2}}{2})}^{2})$

Step 1.2.2.5

Evaluate the exponent.

$v (\frac{25 \cdot 2}{2^{2}} + {(- \frac{5 \sqrt{2}}{2})}^{2})$

Step 1.3

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

$v (\frac{25 \cdot 2}{4} + {(- \frac{5 \sqrt{2}}{2})}^{2})$

Step 1.4

Multiply $25$ by $2$ .

$v (\frac{50}{4} + {(- \frac{5 \sqrt{2}}{2})}^{2})$

Step 1.5

Cancel the common factor of $50$ and $4$ .

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Step 1.5.1

Factor $2$ out of $50$ .

$v (\frac{2 (25)}{4} + {(- \frac{5 \sqrt{2}}{2})}^{2})$

Step 1.5.2

Cancel the common factors.

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Step 1.5.2.1

Factor $2$ out of $4$ .

$v (\frac{2 \cdot 25}{2 \cdot 2} + {(- \frac{5 \sqrt{2}}{2})}^{2})$

Step 1.5.2.2

Cancel the common factor.

$v (\frac{2 \cdot 25}{2 \cdot 2} + {(- \frac{5 \sqrt{2}}{2})}^{2})$

Step 1.5.2.3

Rewrite the expression.

$v (\frac{25}{2} + {(- \frac{5 \sqrt{2}}{2})}^{2})$

Step 1.6

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.6.1

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

$v (\frac{25}{2} + {(- 1)}^{2} {(\frac{5 \sqrt{2}}{2})}^{2})$

Step 1.6.2

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

$v (\frac{25}{2} + {(- 1)}^{2} \frac{{(5 \sqrt{2})}^{2}}{2^{2}})$

Step 1.6.3

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

$v (\frac{25}{2} + {(- 1)}^{2} \frac{5^{2} {\sqrt{2}}^{2}}{2^{2}})$

Step 1.7

Raise $- 1$ to the power of $2$ .

$v (\frac{25}{2} + 1 \frac{5^{2} {\sqrt{2}}^{2}}{2^{2}})$

Step 1.8

Multiply $\frac{5^{2} {\sqrt{2}}^{2}}{2^{2}}$ by $1$ .

$v (\frac{25}{2} + \frac{5^{2} {\sqrt{2}}^{2}}{2^{2}})$

Step 1.9

Simplify the numerator.

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Step 1.9.1

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

$v (\frac{25}{2} + \frac{25 {\sqrt{2}}^{2}}{2^{2}})$

Step 1.9.2

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

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Step 1.9.2.1

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

$v (\frac{25}{2} + \frac{25 {(2^{\frac{1}{2}})}^{2}}{2^{2}})$

Step 1.9.2.2

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

$v (\frac{25}{2} + \frac{25 \cdot 2^{\frac{1}{2} \cdot 2}}{2^{2}})$

Step 1.9.2.3

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

$v (\frac{25}{2} + \frac{25 \cdot 2^{\frac{2}{2}}}{2^{2}})$

Step 1.9.2.4

Cancel the common factor of $2$ .

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Step 1.9.2.4.1

Cancel the common factor.

$v (\frac{25}{2} + \frac{25 \cdot 2^{\frac{2}{2}}}{2^{2}})$

Step 1.9.2.4.2

Rewrite the expression.

$v (\frac{25}{2} + \frac{25 \cdot 2^{1}}{2^{2}})$

Step 1.9.2.5

Evaluate the exponent.

$v (\frac{25}{2} + \frac{25 \cdot 2}{2^{2}})$

Step 1.10

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

$v (\frac{25}{2} + \frac{25 \cdot 2}{4})$

Step 1.11

Multiply $25$ by $2$ .

$v (\frac{25}{2} + \frac{50}{4})$

Step 1.12

Cancel the common factor of $50$ and $4$ .

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Step 1.12.1

Factor $2$ out of $50$ .

$v (\frac{25}{2} + \frac{2 (25)}{4})$

Step 1.12.2

Cancel the common factors.

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Step 1.12.2.1

Factor $2$ out of $4$ .

$v (\frac{25}{2} + \frac{2 \cdot 25}{2 \cdot 2})$

Step 1.12.2.2

Cancel the common factor.

$v (\frac{25}{2} + \frac{2 \cdot 25}{2 \cdot 2})$

Step 1.12.2.3

Rewrite the expression.

$v (\frac{25}{2} + \frac{25}{2})$

Step 2

Combine fractions.

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Step 2.1

Combine the numerators over the common denominator.

$v \frac{25 + 25}{2}$

Step 2.2

Simplify the expression.

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Step 2.2.1

Add $25$ and $25$ .

$v \frac{50}{2}$

Step 2.2.2

Divide $50$ by $2$ .

$v \cdot 25$

Step 2.2.3

Move $25$ to the left of $v$ .

$25 v$