Trigonometry Examples

$\frac{550 + 45 \sqrt{2}}{45 \sqrt{2}}$

Step 1

Cancel the common factor of $550 + 45 \sqrt{2}$ and $45$ .

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Step 1.1

Factor $5$ out of $550$ .

$\frac{5 (110) + 45 \sqrt{2}}{45 \sqrt{2}}$

Step 1.2

Factor $5$ out of $45 \sqrt{2}$ .

$\frac{5 (110) + 5 (9 \sqrt{2})}{45 \sqrt{2}}$

Step 1.3

Factor $5$ out of $5 (110) + 5 (9 \sqrt{2})$ .

$\frac{5 (110 + 9 \sqrt{2})}{45 \sqrt{2}}$

Step 1.4

Cancel the common factors.

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Step 1.4.1

Factor $5$ out of $45 \sqrt{2}$ .

$\frac{5 (110 + 9 \sqrt{2})}{5 (9 \sqrt{2})}$

Step 1.4.2

Cancel the common factor.

$\frac{5 (110 + 9 \sqrt{2})}{5 (9 \sqrt{2})}$

Step 1.4.3

Rewrite the expression.

$\frac{110 + 9 \sqrt{2}}{9 \sqrt{2}}$

Step 2

Multiply $\frac{110 + 9 \sqrt{2}}{9 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{110 + 9 \sqrt{2}}{9 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{110 + 9 \sqrt{2}}{9 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{(110 + 9 \sqrt{2}) \sqrt{2}}{9 \sqrt{2} \sqrt{2}}$

Step 3.2

Move $\sqrt{2}$ .

$\frac{(110 + 9 \sqrt{2}) \sqrt{2}}{9 (\sqrt{2} \sqrt{2})}$

Step 3.3

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

$\frac{(110 + 9 \sqrt{2}) \sqrt{2}}{9 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 3.4

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

$\frac{(110 + 9 \sqrt{2}) \sqrt{2}}{9 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 3.5

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

$\frac{(110 + 9 \sqrt{2}) \sqrt{2}}{9 {\sqrt{2}}^{1 + 1}}$

Step 3.6

Add $1$ and $1$ .

$\frac{(110 + 9 \sqrt{2}) \sqrt{2}}{9 {\sqrt{2}}^{2}}$

Step 3.7

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

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Step 3.7.1

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

$\frac{(110 + 9 \sqrt{2}) \sqrt{2}}{9 {(2^{\frac{1}{2}})}^{2}}$

Step 3.7.2

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

$\frac{(110 + 9 \sqrt{2}) \sqrt{2}}{9 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 3.7.3

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

$\frac{(110 + 9 \sqrt{2}) \sqrt{2}}{9 \cdot 2^{\frac{2}{2}}}$

Step 3.7.4

Cancel the common factor of $2$ .

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Step 3.7.4.1

Cancel the common factor.

$\frac{(110 + 9 \sqrt{2}) \sqrt{2}}{9 \cdot 2^{\frac{2}{2}}}$

Step 3.7.4.2

Rewrite the expression.

$\frac{(110 + 9 \sqrt{2}) \sqrt{2}}{9 \cdot 2^{1}}$

Step 3.7.5

Evaluate the exponent.

$\frac{(110 + 9 \sqrt{2}) \sqrt{2}}{9 \cdot 2}$

Step 4

Multiply $9$ by $2$ .

$\frac{(110 + 9 \sqrt{2}) \sqrt{2}}{18}$

Step 5

Apply the distributive property.

$\frac{110 \sqrt{2} + 9 \sqrt{2} \sqrt{2}}{18}$

Step 6

Multiply $9 \sqrt{2} \sqrt{2}$ .

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Step 6.1

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

$\frac{110 \sqrt{2} + 9 ({\sqrt{2}}^{1} \sqrt{2})}{18}$

Step 6.2

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

$\frac{110 \sqrt{2} + 9 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{18}$

Step 6.3

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

$\frac{110 \sqrt{2} + 9 {\sqrt{2}}^{1 + 1}}{18}$

Step 6.4

Add $1$ and $1$ .

$\frac{110 \sqrt{2} + 9 {\sqrt{2}}^{2}}{18}$

Step 7

Simplify each term.

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Step 7.1

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

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Step 7.1.1

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

$\frac{110 \sqrt{2} + 9 {(2^{\frac{1}{2}})}^{2}}{18}$

Step 7.1.2

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

$\frac{110 \sqrt{2} + 9 \cdot 2^{\frac{1}{2} \cdot 2}}{18}$

Step 7.1.3

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

$\frac{110 \sqrt{2} + 9 \cdot 2^{\frac{2}{2}}}{18}$

Step 7.1.4

Cancel the common factor of $2$ .

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Step 7.1.4.1

Cancel the common factor.

$\frac{110 \sqrt{2} + 9 \cdot 2^{\frac{2}{2}}}{18}$

Step 7.1.4.2

Rewrite the expression.

$\frac{110 \sqrt{2} + 9 \cdot 2^{1}}{18}$

Step 7.1.5

Evaluate the exponent.

$\frac{110 \sqrt{2} + 9 \cdot 2}{18}$

Step 7.2

Multiply $9$ by $2$ .

$\frac{110 \sqrt{2} + 18}{18}$

Step 8

Cancel the common factor of $110 \sqrt{2} + 18$ and $18$ .

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Step 8.1

Factor $2$ out of $110 \sqrt{2}$ .

$\frac{2 (55 \sqrt{2}) + 18}{18}$

Step 8.2

Factor $2$ out of $18$ .

$\frac{2 (55 \sqrt{2}) + 2 (9)}{18}$

Step 8.3

Factor $2$ out of $2 (55 \sqrt{2}) + 2 (9)$ .

$\frac{2 (55 \sqrt{2} + 9)}{18}$

Step 8.4

Cancel the common factors.

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Step 8.4.1

Factor $2$ out of $18$ .

$\frac{2 (55 \sqrt{2} + 9)}{2 (9)}$

Step 8.4.2

Cancel the common factor.

$\frac{2 (55 \sqrt{2} + 9)}{2 \cdot 9}$

Step 8.4.3

Rewrite the expression.

$\frac{55 \sqrt{2} + 9}{9}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{55 \sqrt{2} + 9}{9}$

Decimal Form:

$9.64241621 \dots$