Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

$\frac{\sqrt{3}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} + \sqrt{24}$

Step 1.3

Combine and simplify the denominator.

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Step 1.3.1

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

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

Step 1.3.2

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

$\frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} + \sqrt{24}$

Step 1.3.3

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

$\frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} + \sqrt{24}$

Step 1.3.4

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

$\frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1 + 1}} + \sqrt{24}$

Step 1.3.5

Add $1$ and $1$ .

$\frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{2}} + \sqrt{24}$

Step 1.3.6

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

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Step 1.3.6.1

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

$\frac{\sqrt{3} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} + \sqrt{24}$

Step 1.3.6.2

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

$\frac{\sqrt{3} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}} + \sqrt{24}$

Step 1.3.6.3

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

$\frac{\sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}} + \sqrt{24}$

Step 1.3.6.4

Cancel the common factor of $2$ .

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Step 1.3.6.4.1

Cancel the common factor.

$\frac{\sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}} + \sqrt{24}$

Step 1.3.6.4.2

Rewrite the expression.

$\frac{\sqrt{3} \sqrt{2}}{2^{1}} + \sqrt{24}$

Step 1.3.6.5

Evaluate the exponent.

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

Step 1.4

Simplify the numerator.

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Step 1.4.1

Combine using the product rule for radicals.

$\frac{\sqrt{3 \cdot 2}}{2} + \sqrt{24}$

Step 1.4.2

Multiply $3$ by $2$ .

$\frac{\sqrt{6}}{2} + \sqrt{24}$

Step 1.5

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 1.5.1

Factor $4$ out of $24$ .

$\frac{\sqrt{6}}{2} + \sqrt{4 (6)}$

Step 1.5.2

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

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

Step 1.6

Pull terms out from under the radical.

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

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

Add $\sqrt{6}$ and $4 \sqrt{6}$ .

$\frac{5 \sqrt{6}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{5 \sqrt{6}}{2}$

Decimal Form:

$6.12372435 \dots$