Trigonometry Examples

$\sqrt{\frac{35^{2}}{2 {(35)}^{2} + 64 (7)}}$

Step 1

Simplify the expression.

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Step 1.1

Raise $35$ to the power of $2$ .

$\sqrt{\frac{35^{2}}{2 \cdot 1225 + 64 (7)}}$

Step 1.2

Multiply $2$ by $1225$ .

$\sqrt{\frac{35^{2}}{2450 + 64 (7)}}$

Step 1.3

Multiply $64$ by $7$ .

$\sqrt{\frac{35^{2}}{2450 + 448}}$

Step 1.4

Add $2450$ and $448$ .

$\sqrt{\frac{35^{2}}{2898}}$

Step 1.5

Raise $35$ to the power of $2$ .

$\sqrt{\frac{1225}{2898}}$

Step 2

Cancel the common factor of $1225$ and $2898$ .

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Step 2.1

Factor $7$ out of $1225$ .

$\sqrt{\frac{7 (175)}{2898}}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $7$ out of $2898$ .

$\sqrt{\frac{7 \cdot 175}{7 \cdot 414}}$

Step 2.2.2

Cancel the common factor.

$\sqrt{\frac{7 \cdot 175}{7 \cdot 414}}$

Step 2.2.3

Rewrite the expression.

$\sqrt{\frac{175}{414}}$

Step 3

Rewrite $\sqrt{\frac{175}{414}}$ as $\frac{\sqrt{175}}{\sqrt{414}}$ .

$\frac{\sqrt{175}}{\sqrt{414}}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $175$ as $5^{2} \cdot 7$ .

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Step 4.1.1

Factor $25$ out of $175$ .

$\frac{\sqrt{25 (7)}}{\sqrt{414}}$

Step 4.1.2

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

$\frac{\sqrt{5^{2} \cdot 7}}{\sqrt{414}}$

Step 4.2

Pull terms out from under the radical.

$\frac{5 \sqrt{7}}{\sqrt{414}}$

Step 5

Simplify the denominator.

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Step 5.1

Rewrite $414$ as $3^{2} \cdot 46$ .

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Step 5.1.1

Factor $9$ out of $414$ .

$\frac{5 \sqrt{7}}{\sqrt{9 (46)}}$

Step 5.1.2

Rewrite $9$ as $3^{2}$ .

$\frac{5 \sqrt{7}}{\sqrt{3^{2} \cdot 46}}$

Step 5.2

Pull terms out from under the radical.

$\frac{5 \sqrt{7}}{3 \sqrt{46}}$

Step 6

Multiply $\frac{5 \sqrt{7}}{3 \sqrt{46}}$ by $\frac{\sqrt{46}}{\sqrt{46}}$ .

$\frac{5 \sqrt{7}}{3 \sqrt{46}} \cdot \frac{\sqrt{46}}{\sqrt{46}}$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{5 \sqrt{7}}{3 \sqrt{46}}$ by $\frac{\sqrt{46}}{\sqrt{46}}$ .

$\frac{5 \sqrt{7} \sqrt{46}}{3 \sqrt{46} \sqrt{46}}$

Step 7.2

Move $\sqrt{46}$ .

$\frac{5 \sqrt{7} \sqrt{46}}{3 (\sqrt{46} \sqrt{46})}$

Step 7.3

Raise $\sqrt{46}$ to the power of $1$ .

$\frac{5 \sqrt{7} \sqrt{46}}{3 ({\sqrt{46}}^{1} \sqrt{46})}$

Step 7.4

Raise $\sqrt{46}$ to the power of $1$ .

$\frac{5 \sqrt{7} \sqrt{46}}{3 ({\sqrt{46}}^{1} {\sqrt{46}}^{1})}$

Step 7.5

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

$\frac{5 \sqrt{7} \sqrt{46}}{3 {\sqrt{46}}^{1 + 1}}$

Step 7.6

Add $1$ and $1$ .

$\frac{5 \sqrt{7} \sqrt{46}}{3 {\sqrt{46}}^{2}}$

Step 7.7

Rewrite ${\sqrt{46}}^{2}$ as $46$ .

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Step 7.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{46}$ as $46^{\frac{1}{2}}$ .

$\frac{5 \sqrt{7} \sqrt{46}}{3 {(46^{\frac{1}{2}})}^{2}}$

Step 7.7.2

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

$\frac{5 \sqrt{7} \sqrt{46}}{3 \cdot 46^{\frac{1}{2} \cdot 2}}$

Step 7.7.3

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

$\frac{5 \sqrt{7} \sqrt{46}}{3 \cdot 46^{\frac{2}{2}}}$

Step 7.7.4

Cancel the common factor of $2$ .

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Step 7.7.4.1

Cancel the common factor.

$\frac{5 \sqrt{7} \sqrt{46}}{3 \cdot 46^{\frac{2}{2}}}$

Step 7.7.4.2

Rewrite the expression.

$\frac{5 \sqrt{7} \sqrt{46}}{3 \cdot 46^{1}}$

Step 7.7.5

Evaluate the exponent.

$\frac{5 \sqrt{7} \sqrt{46}}{3 \cdot 46}$

Step 8

Simplify the numerator.

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Step 8.1

Combine using the product rule for radicals.

$\frac{5 \sqrt{46 \cdot 7}}{3 \cdot 46}$

Step 8.2

Multiply $46$ by $7$ .

$\frac{5 \sqrt{322}}{3 \cdot 46}$

Step 9

Multiply $3$ by $46$ .

$\frac{5 \sqrt{322}}{138}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{5 \sqrt{322}}{138}$

Decimal Form:

$0.65015791 \dots$