Trigonometry Examples

$\frac{3 \sqrt{72}}{2 \sqrt{20}}$

Step 1

Combine $\sqrt{72}$ and $\sqrt{20}$ into a single radical.

$\frac{3 \sqrt{\frac{72}{20}}}{2}$

Step 2

Cancel the common factor of $72$ and $20$ .

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Step 2.1

Factor $4$ out of $72$ .

$\frac{3 \sqrt{\frac{4 (18)}{20}}}{2}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $4$ out of $20$ .

$\frac{3 \sqrt{\frac{4 \cdot 18}{4 \cdot 5}}}{2}$

Step 2.2.2

Cancel the common factor.

$\frac{3 \sqrt{\frac{4 \cdot 18}{4 \cdot 5}}}{2}$

Step 2.2.3

Rewrite the expression.

$\frac{3 \sqrt{\frac{18}{5}}}{2}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $\sqrt{\frac{18}{5}}$ as $\frac{\sqrt{18}}{\sqrt{5}}$ .

$\frac{3 \frac{\sqrt{18}}{\sqrt{5}}}{2}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 3.2.1.1

Factor $9$ out of $18$ .

$\frac{3 \frac{\sqrt{9 (2)}}{\sqrt{5}}}{2}$

Step 3.2.1.2

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

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

Step 3.2.2

Pull terms out from under the radical.

$\frac{3 \frac{3 \sqrt{2}}{\sqrt{5}}}{2}$

Step 3.3

Multiply $\frac{3 \sqrt{2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{3 (\frac{3 \sqrt{2}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}})}{2}$

Step 3.4

Combine and simplify the denominator.

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Step 3.4.1

Multiply $\frac{3 \sqrt{2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{3 \frac{3 \sqrt{2} \sqrt{5}}{\sqrt{5} \sqrt{5}}}{2}$

Step 3.4.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{3 \frac{3 \sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}}{2}$

Step 3.4.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{3 \frac{3 \sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}}{2}$

Step 3.4.4

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

$\frac{3 \frac{3 \sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}}{2}$

Step 3.4.5

Add $1$ and $1$ .

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

Step 3.4.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 3.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

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

Step 3.4.6.2

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

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

Step 3.4.6.3

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

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

Step 3.4.6.4

Cancel the common factor of $2$ .

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Step 3.4.6.4.1

Cancel the common factor.

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

Step 3.4.6.4.2

Rewrite the expression.

$\frac{3 \frac{3 \sqrt{2} \sqrt{5}}{5^{1}}}{2}$

Step 3.4.6.5

Evaluate the exponent.

$\frac{3 \frac{3 \sqrt{2} \sqrt{5}}{5}}{2}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Combine using the product rule for radicals.

$\frac{3 \frac{3 \sqrt{5 \cdot 2}}{5}}{2}$

Step 3.5.2

Multiply $5$ by $2$ .

$\frac{3 \frac{3 \sqrt{10}}{5}}{2}$

Step 4

Combine $3$ and $\frac{3 \sqrt{10}}{5}$ .

$\frac{\frac{3 (3 \sqrt{10})}{5}}{2}$

Step 5

Multiply $3$ by $3$ .

$\frac{\frac{9 \sqrt{10}}{5}}{2}$

Step 6

Multiply the numerator by the reciprocal of the denominator.

$\frac{9 \sqrt{10}}{5} \cdot \frac{1}{2}$

Step 7

Multiply $\frac{9 \sqrt{10}}{5} \cdot \frac{1}{2}$ .

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Step 7.1

Multiply $\frac{9 \sqrt{10}}{5}$ by $\frac{1}{2}$ .

$\frac{9 \sqrt{10}}{5 \cdot 2}$

Step 7.2

Multiply $5$ by $2$ .

$\frac{9 \sqrt{10}}{10}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{9 \sqrt{10}}{10}$

Decimal Form:

$2.84604989 \dots$