Trigonometry Examples

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

Step 1

Multiply $\frac{4 \sqrt{3}}{3 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 2

Simplify terms.

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Step 2.1

Combine and simplify the denominator.

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Step 2.1.1

Multiply $\frac{4 \sqrt{3}}{3 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 2.1.2

Move $\sqrt{2}$ .

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

Step 2.1.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{4 \sqrt{3} \sqrt{2}}{3 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 2.1.4

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{4 \sqrt{3} \sqrt{2}}{3 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 2.1.5

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

$\frac{4 \sqrt{3} \sqrt{2}}{3 {\sqrt{2}}^{1 + 1}}$

Step 2.1.6

Add $1$ and $1$ .

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

Step 2.1.7

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

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Step 2.1.7.1

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

$\frac{4 \sqrt{3} \sqrt{2}}{3 {(2^{\frac{1}{2}})}^{2}}$

Step 2.1.7.2

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

$\frac{4 \sqrt{3} \sqrt{2}}{3 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 2.1.7.3

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

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

Step 2.1.7.4

Cancel the common factor of $2$ .

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Step 2.1.7.4.1

Cancel the common factor.

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

Step 2.1.7.4.2

Rewrite the expression.

$\frac{4 \sqrt{3} \sqrt{2}}{3 \cdot 2^{1}}$

Step 2.1.7.5

Evaluate the exponent.

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

Step 2.2

Cancel the common factor of $4$ and $2$ .

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Step 2.2.1

Factor $2$ out of $4 \sqrt{3} \sqrt{2}$ .

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

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Factor $2$ out of $3 \cdot 2$ .

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

Step 2.2.2.2

Cancel the common factor.

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

Step 2.2.2.3

Rewrite the expression.

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

Step 3

Simplify the numerator.

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Step 3.1

Combine using the product rule for radicals.

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

Step 3.2

Multiply $2$ by $3$ .

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.63299316 \dots$