Trigonometry Examples

$\frac{6 \sqrt{32}}{4 \sqrt{24}}$

Step 1

Cancel the common factor of $6$ and $4$ .

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Step 1.1

Factor $2$ out of $6 \sqrt{32}$ .

$\frac{2 (3 \sqrt{32})}{4 \sqrt{24}}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $2$ out of $4 \sqrt{24}$ .

$\frac{2 (3 \sqrt{32})}{2 (2 \sqrt{24})}$

Step 1.2.2

Cancel the common factor.

$\frac{2 (3 \sqrt{32})}{2 (2 \sqrt{24})}$

Step 1.2.3

Rewrite the expression.

$\frac{3 \sqrt{32}}{2 \sqrt{24}}$

Step 2

Combine $\sqrt{32}$ and $\sqrt{24}$ into a single radical.

$\frac{3 \sqrt{\frac{32}{24}}}{2}$

Step 3

Cancel the common factor of $32$ and $24$ .

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Step 3.1

Factor $8$ out of $32$ .

$\frac{3 \sqrt{\frac{8 (4)}{24}}}{2}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $8$ out of $24$ .

$\frac{3 \sqrt{\frac{8 \cdot 4}{8 \cdot 3}}}{2}$

Step 3.2.2

Cancel the common factor.

$\frac{3 \sqrt{\frac{8 \cdot 4}{8 \cdot 3}}}{2}$

Step 3.2.3

Rewrite the expression.

$\frac{3 \sqrt{\frac{4}{3}}}{2}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $\sqrt{\frac{4}{3}}$ as $\frac{\sqrt{4}}{\sqrt{3}}$ .

$\frac{3 \frac{\sqrt{4}}{\sqrt{3}}}{2}$

Step 4.2

Simplify the numerator.

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Step 4.2.1

Rewrite $4$ as $2^{2}$ .

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

Step 4.2.2

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

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

Step 4.3

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

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

Step 4.4

Combine and simplify the denominator.

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Step 4.4.1

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

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

Step 4.4.2

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 4.4.3

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 4.4.4

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

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

Step 4.4.5

Add $1$ and $1$ .

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

Step 4.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 4.4.6.2

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

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

Step 4.4.6.3

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

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

Step 4.4.6.4

Cancel the common factor of $2$ .

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Step 4.4.6.4.1

Cancel the common factor.

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

Step 4.4.6.4.2

Rewrite the expression.

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

Step 4.4.6.5

Evaluate the exponent.

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

Step 5

Combine $3$ and $\frac{2 \sqrt{3}}{3}$ .

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

Step 6

Multiply $3$ by $2$ .

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

Step 7

Reduce the expression by cancelling the common factors.

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Step 7.1

Reduce the expression $\frac{6 \sqrt{3}}{3}$ by cancelling the common factors.

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Step 7.1.1

Factor $3$ out of $6 \sqrt{3}$ .

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

Step 7.1.2

Factor $3$ out of $3$ .

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

Step 7.1.3

Cancel the common factor.

$\frac{\frac{3 (2 \sqrt{3})}{3 \cdot 1}}{2}$

Step 7.1.4

Rewrite the expression.

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

Step 7.2

Divide $2 \sqrt{3}$ by $1$ .

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

Step 8

Cancel the common factor of $2$ .

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Step 8.1

Cancel the common factor.

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

Step 8.2

Divide $\sqrt{3}$ by $1$ .

$\sqrt{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\sqrt{3}$

Decimal Form:

$1.73205080 \dots$