Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

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

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

Step 1.2.2

Move $\sqrt{3}$ .

$\frac{3 \sqrt{2} \sqrt{3}}{2 (\sqrt{3} \sqrt{3})} - 2 \sqrt{6}$

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

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

$\frac{3 \sqrt{2} \sqrt{3}}{2 {\sqrt{3}}^{1 + 1}} - 2 \sqrt{6}$

Step 1.2.6

Add $1$ and $1$ .

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

Step 1.2.7

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

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Step 1.2.7.1

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

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

Step 1.2.7.2

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

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

Step 1.2.7.3

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

$\frac{3 \sqrt{2} \sqrt{3}}{2 \cdot 3^{\frac{2}{2}}} - 2 \sqrt{6}$

Step 1.2.7.4

Cancel the common factor of $2$ .

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Step 1.2.7.4.1

Cancel the common factor.

$\frac{3 \sqrt{2} \sqrt{3}}{2 \cdot 3^{\frac{2}{2}}} - 2 \sqrt{6}$

Step 1.2.7.4.2

Rewrite the expression.

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

Step 1.2.7.5

Evaluate the exponent.

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

Step 1.3

Cancel the common factor of $3$ .

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Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

Rewrite the expression.

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

Step 1.4

Simplify the numerator.

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Step 1.4.1

Combine using the product rule for radicals.

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

Step 1.4.2

Multiply $2$ by $3$ .

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

Step 2

To write $- 2 \sqrt{6}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\sqrt{6}}{2} - 2 \sqrt{6} \cdot \frac{2}{2}$

Step 3

Combine fractions.

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Step 3.1

Combine $- 2 \sqrt{6}$ and $\frac{2}{2}$ .

$\frac{\sqrt{6}}{2} + \frac{- 2 \sqrt{6} \cdot 2}{2}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{\sqrt{6} - 2 \sqrt{6} \cdot 2}{2}$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $2$ by $- 2$ .

$\frac{\sqrt{6} - 4 \sqrt{6}}{2}$

Step 4.2

Subtract $4 \sqrt{6}$ from $\sqrt{6}$ .

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

Step 5

Move the negative in front of the fraction.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 3.67423461 \dots$