Trigonometry Examples

$\sqrt{\frac{189}{34}}$

Step 1

Rewrite $\sqrt{\frac{189}{34}}$ as $\frac{\sqrt{189}}{\sqrt{34}}$ .

$\frac{\sqrt{189}}{\sqrt{34}}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $189$ as $3^{2} \cdot 21$ .

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Step 2.1.1

Factor $9$ out of $189$ .

$\frac{\sqrt{9 (21)}}{\sqrt{34}}$

Step 2.1.2

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

$\frac{\sqrt{3^{2} \cdot 21}}{\sqrt{34}}$

Step 2.2

Pull terms out from under the radical.

$\frac{3 \sqrt{21}}{\sqrt{34}}$

Step 3

Multiply $\frac{3 \sqrt{21}}{\sqrt{34}}$ by $\frac{\sqrt{34}}{\sqrt{34}}$ .

$\frac{3 \sqrt{21}}{\sqrt{34}} \cdot \frac{\sqrt{34}}{\sqrt{34}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{3 \sqrt{21}}{\sqrt{34}}$ by $\frac{\sqrt{34}}{\sqrt{34}}$ .

$\frac{3 \sqrt{21} \sqrt{34}}{\sqrt{34} \sqrt{34}}$

Step 4.2

Raise $\sqrt{34}$ to the power of $1$ .

$\frac{3 \sqrt{21} \sqrt{34}}{{\sqrt{34}}^{1} \sqrt{34}}$

Step 4.3

Raise $\sqrt{34}$ to the power of $1$ .

$\frac{3 \sqrt{21} \sqrt{34}}{{\sqrt{34}}^{1} {\sqrt{34}}^{1}}$

Step 4.4

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

$\frac{3 \sqrt{21} \sqrt{34}}{{\sqrt{34}}^{1 + 1}}$

Step 4.5

Add $1$ and $1$ .

$\frac{3 \sqrt{21} \sqrt{34}}{{\sqrt{34}}^{2}}$

Step 4.6

Rewrite ${\sqrt{34}}^{2}$ as $34$ .

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Step 4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{34}$ as $34^{\frac{1}{2}}$ .

$\frac{3 \sqrt{21} \sqrt{34}}{{(34^{\frac{1}{2}})}^{2}}$

Step 4.6.2

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

$\frac{3 \sqrt{21} \sqrt{34}}{34^{\frac{1}{2} \cdot 2}}$

Step 4.6.3

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

$\frac{3 \sqrt{21} \sqrt{34}}{34^{\frac{2}{2}}}$

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

$\frac{3 \sqrt{21} \sqrt{34}}{34^{\frac{2}{2}}}$

Step 4.6.4.2

Rewrite the expression.

$\frac{3 \sqrt{21} \sqrt{34}}{34^{1}}$

Step 4.6.5

Evaluate the exponent.

$\frac{3 \sqrt{21} \sqrt{34}}{34}$

Step 5

Simplify the numerator.

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Step 5.1

Combine using the product rule for radicals.

$\frac{3 \sqrt{34 \cdot 21}}{34}$

Step 5.2

Multiply $34$ by $21$ .

$\frac{3 \sqrt{714}}{34}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{3 \sqrt{714}}{34}$

Decimal Form:

$2.35771574 \dots$