Trigonometry Examples

$\frac{6}{\frac{7}{\sqrt{\frac{3}{7}}}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$6 \frac{\sqrt{\frac{3}{7}}}{7}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $\sqrt{\frac{3}{7}}$ as $\frac{\sqrt{3}}{\sqrt{7}}$ .

$6 \frac{\frac{\sqrt{3}}{\sqrt{7}}}{7}$

Step 2.2

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

$6 \frac{\frac{\sqrt{3}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}}{7}$

Step 2.3

Combine and simplify the denominator.

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Step 2.3.1

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

$6 \frac{\frac{\sqrt{3} \sqrt{7}}{\sqrt{7} \sqrt{7}}}{7}$

Step 2.3.2

Raise $\sqrt{7}$ to the power of $1$ .

$6 \frac{\frac{\sqrt{3} \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}}{7}$

Step 2.3.3

Raise $\sqrt{7}$ to the power of $1$ .

$6 \frac{\frac{\sqrt{3} \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}}{7}$

Step 2.3.4

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

$6 \frac{\frac{\sqrt{3} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}}{7}$

Step 2.3.5

Add $1$ and $1$ .

$6 \frac{\frac{\sqrt{3} \sqrt{7}}{{\sqrt{7}}^{2}}}{7}$

Step 2.3.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 2.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$6 \frac{\frac{\sqrt{3} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}}{7}$

Step 2.3.6.2

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

$6 \frac{\frac{\sqrt{3} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}}{7}$

Step 2.3.6.3

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

$6 \frac{\frac{\sqrt{3} \sqrt{7}}{7^{\frac{2}{2}}}}{7}$

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.

$6 \frac{\frac{\sqrt{3} \sqrt{7}}{7^{\frac{2}{2}}}}{7}$

Step 2.3.6.4.2

Rewrite the expression.

$6 \frac{\frac{\sqrt{3} \sqrt{7}}{7^{1}}}{7}$

Step 2.3.6.5

Evaluate the exponent.

$6 \frac{\frac{\sqrt{3} \sqrt{7}}{7}}{7}$

Step 2.4

Simplify the numerator.

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Step 2.4.1

Combine using the product rule for radicals.

$6 \frac{\frac{\sqrt{3 \cdot 7}}{7}}{7}$

Step 2.4.2

Multiply $3$ by $7$ .

$6 \frac{\frac{\sqrt{21}}{7}}{7}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$6 (\frac{\sqrt{21}}{7} \cdot \frac{1}{7})$

Step 4

Multiply $\frac{\sqrt{21}}{7} \cdot \frac{1}{7}$ .

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Step 4.1

Multiply $\frac{\sqrt{21}}{7}$ by $\frac{1}{7}$ .

$6 \frac{\sqrt{21}}{7 \cdot 7}$

Step 4.2

Multiply $7$ by $7$ .

$6 \frac{\sqrt{21}}{49}$

Step 5

Combine $6$ and $\frac{\sqrt{21}}{49}$ .

$\frac{6 \sqrt{21}}{49}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{6 \sqrt{21}}{49}$

Decimal Form:

$0.56113171 \dots$