Trigonometry Examples

$\sqrt{\frac{7}{5} \cdot \sqrt{\frac{3}{14}}}$

Step 1

Combine $\frac{7}{5}$ and $\sqrt{\frac{3}{14}}$ .

$\sqrt{\frac{7 \sqrt{\frac{3}{14}}}{5}}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $\sqrt{\frac{3}{14}}$ as $\frac{\sqrt{3}}{\sqrt{14}}$ .

$\sqrt{\frac{7 \frac{\sqrt{3}}{\sqrt{14}}}{5}}$

Step 2.2

Multiply $\frac{\sqrt{3}}{\sqrt{14}}$ by $\frac{\sqrt{14}}{\sqrt{14}}$ .

$\sqrt{\frac{7 (\frac{\sqrt{3}}{\sqrt{14}} \cdot \frac{\sqrt{14}}{\sqrt{14}})}{5}}$

Step 2.3

Combine and simplify the denominator.

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Step 2.3.1

Multiply $\frac{\sqrt{3}}{\sqrt{14}}$ by $\frac{\sqrt{14}}{\sqrt{14}}$ .

$\sqrt{\frac{7 \frac{\sqrt{3} \sqrt{14}}{\sqrt{14} \sqrt{14}}}{5}}$

Step 2.3.2

Raise $\sqrt{14}$ to the power of $1$ .

$\sqrt{\frac{7 \frac{\sqrt{3} \sqrt{14}}{{\sqrt{14}}^{1} \sqrt{14}}}{5}}$

Step 2.3.3

Raise $\sqrt{14}$ to the power of $1$ .

$\sqrt{\frac{7 \frac{\sqrt{3} \sqrt{14}}{{\sqrt{14}}^{1} {\sqrt{14}}^{1}}}{5}}$

Step 2.3.4

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

$\sqrt{\frac{7 \frac{\sqrt{3} \sqrt{14}}{{\sqrt{14}}^{1 + 1}}}{5}}$

Step 2.3.5

Add $1$ and $1$ .

$\sqrt{\frac{7 \frac{\sqrt{3} \sqrt{14}}{{\sqrt{14}}^{2}}}{5}}$

Step 2.3.6

Rewrite ${\sqrt{14}}^{2}$ as $14$ .

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Step 2.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{14}$ as $14^{\frac{1}{2}}$ .

$\sqrt{\frac{7 \frac{\sqrt{3} \sqrt{14}}{{(14^{\frac{1}{2}})}^{2}}}{5}}$

Step 2.3.6.2

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

$\sqrt{\frac{7 \frac{\sqrt{3} \sqrt{14}}{14^{\frac{1}{2} \cdot 2}}}{5}}$

Step 2.3.6.3

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

$\sqrt{\frac{7 \frac{\sqrt{3} \sqrt{14}}{14^{\frac{2}{2}}}}{5}}$

Step 2.3.6.4

Cancel the common factor of $2$ .

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Step 2.3.6.4.1

Cancel the common factor.

$\sqrt{\frac{7 \frac{\sqrt{3} \sqrt{14}}{14^{\frac{2}{2}}}}{5}}$

Step 2.3.6.4.2

Rewrite the expression.

$\sqrt{\frac{7 \frac{\sqrt{3} \sqrt{14}}{14^{1}}}{5}}$

Step 2.3.6.5

Evaluate the exponent.

$\sqrt{\frac{7 \frac{\sqrt{3} \sqrt{14}}{14}}{5}}$

Step 2.4

Simplify the numerator.

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Step 2.4.1

Combine using the product rule for radicals.

$\sqrt{\frac{7 \frac{\sqrt{3 \cdot 14}}{14}}{5}}$

Step 2.4.2

Multiply $3$ by $14$ .

$\sqrt{\frac{7 \frac{\sqrt{42}}{14}}{5}}$

Step 3

Simplify terms.

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Step 3.1

Combine $7$ and $\frac{\sqrt{42}}{14}$ .

$\sqrt{\frac{\frac{7 \sqrt{42}}{14}}{5}}$

Step 3.2

Reduce the expression $\frac{7 \sqrt{42}}{14}$ by cancelling the common factors.

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Step 3.2.1

Factor $7$ out of $7 \sqrt{42}$ .

$\sqrt{\frac{\frac{7 (\sqrt{42})}{14}}{5}}$

Step 3.2.2

Factor $7$ out of $14$ .

$\sqrt{\frac{\frac{7 \sqrt{42}}{7 \cdot 2}}{5}}$

Step 3.2.3

Cancel the common factor.

$\sqrt{\frac{\frac{7 \sqrt{42}}{7 \cdot 2}}{5}}$

Step 3.2.4

Rewrite the expression.

$\sqrt{\frac{\frac{\sqrt{42}}{2}}{5}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{\sqrt{42}}{2} \cdot \frac{1}{5}}$

Step 5

Multiply $\frac{\sqrt{42}}{2} \cdot \frac{1}{5}$ .

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Step 5.1

Multiply $\frac{\sqrt{42}}{2}$ by $\frac{1}{5}$ .

$\sqrt{\frac{\sqrt{42}}{2 \cdot 5}}$

Step 5.2

Multiply $2$ by $5$ .

$\sqrt{\frac{\sqrt{42}}{10}}$

Step 6

Rewrite $\sqrt{\frac{\sqrt{42}}{10}}$ as $\frac{\sqrt{\sqrt{42}}}{\sqrt{10}}$ .

$\frac{\sqrt{\sqrt{42}}}{\sqrt{10}}$

Step 7

Rewrite $\sqrt{\sqrt{42}}$ as $\sqrt[4]{42}$ .

$\frac{\sqrt[4]{42}}{\sqrt{10}}$

Step 8

Multiply $\frac{\sqrt[4]{42}}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\frac{\sqrt[4]{42}}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}}$

Step 9

Combine and simplify the denominator.

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Step 9.1

Multiply $\frac{\sqrt[4]{42}}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\frac{\sqrt[4]{42} \sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 9.2

Raise $\sqrt{10}$ to the power of $1$ .

$\frac{\sqrt[4]{42} \sqrt{10}}{{\sqrt{10}}^{1} \sqrt{10}}$

Step 9.3

Raise $\sqrt{10}$ to the power of $1$ .

$\frac{\sqrt[4]{42} \sqrt{10}}{{\sqrt{10}}^{1} {\sqrt{10}}^{1}}$

Step 9.4

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

$\frac{\sqrt[4]{42} \sqrt{10}}{{\sqrt{10}}^{1 + 1}}$

Step 9.5

Add $1$ and $1$ .

$\frac{\sqrt[4]{42} \sqrt{10}}{{\sqrt{10}}^{2}}$

Step 9.6

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 9.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\frac{\sqrt[4]{42} \sqrt{10}}{{(10^{\frac{1}{2}})}^{2}}$

Step 9.6.2

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

$\frac{\sqrt[4]{42} \sqrt{10}}{10^{\frac{1}{2} \cdot 2}}$

Step 9.6.3

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

$\frac{\sqrt[4]{42} \sqrt{10}}{10^{\frac{2}{2}}}$

Step 9.6.4

Cancel the common factor of $2$ .

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Step 9.6.4.1

Cancel the common factor.

$\frac{\sqrt[4]{42} \sqrt{10}}{10^{\frac{2}{2}}}$

Step 9.6.4.2

Rewrite the expression.

$\frac{\sqrt[4]{42} \sqrt{10}}{10^{1}}$

Step 9.6.5

Evaluate the exponent.

$\frac{\sqrt[4]{42} \sqrt{10}}{10}$

Step 10

Simplify the numerator.

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Step 10.1

Rewrite the expression using the least common index of $4$ .

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Step 10.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\frac{\sqrt[4]{42} \cdot 10^{\frac{1}{2}}}{10}$

Step 10.1.2

Rewrite $10^{\frac{1}{2}}$ as $10^{\frac{2}{4}}$ .

$\frac{\sqrt[4]{42} \cdot 10^{\frac{2}{4}}}{10}$

Step 10.1.3

Rewrite $10^{\frac{2}{4}}$ as $\sqrt[4]{10^{2}}$ .

$\frac{\sqrt[4]{42} \sqrt[4]{10^{2}}}{10}$

Step 10.2

Combine using the product rule for radicals.

$\frac{\sqrt[4]{42 \cdot 10^{2}}}{10}$

Step 11

Simplify the numerator.

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Step 11.1

Rewrite $42 \cdot 10^{2}$ as $0.42 \cdot 10^{4}$ .

$\frac{\sqrt[4]{0.42 \cdot 10^{4}}}{10}$

Step 11.2

Rewrite $\sqrt[4]{0.42 \cdot 10^{4}}$ as $\sqrt[4]{0.42} \cdot \sqrt[4]{10^{4}}$ .

$\frac{\sqrt[4]{0.42} \cdot \sqrt[4]{10^{4}}}{10}$

Step 11.3

Evaluate the root.

$\frac{0.80503047 \cdot \sqrt[4]{10^{4}}}{10}$

Step 11.4

Pull terms out from under the radical, assuming positive real numbers.

$\frac{0.80503047 \cdot 10}{10}$

Step 11.5

Raise $10$ to the power of $1$ .

$\frac{0.80503047 \cdot 10^{1}}{10}$

Step 11.6

Move the decimal point in $0.80503047$ to the right by $1$ place and decrease the power of $10^{1}$ by $1$ .

$\frac{8.05030477 \cdot 10^{0}}{10}$

Step 11.7

Convert $8.05030477 \cdot 10^{0}$ from scientific notation.

$\frac{8.05030477}{10}$

Step 12

Divide $8.05030477$ by $10$ .

$0.80503047$