Basic Math Examples

$\frac{\sqrt{6 \sqrt{24}}}{\sqrt{8}}$

Step 1

Combine $\sqrt{6 \sqrt{24}}$ and $\sqrt{8}$ into a single radical.

$\sqrt{\frac{6 \sqrt{24}}{8}}$

Step 2

Reduce the expression $\frac{6 \sqrt{24}}{8}$ by cancelling the common factors.

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Step 2.1

Factor $2$ out of $6 \sqrt{24}$ .

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

Step 2.2

Factor $2$ out of $8$ .

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

Step 2.3

Cancel the common factor.

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

Step 2.4

Rewrite the expression.

$\sqrt{\frac{3 \sqrt{24}}{4}}$

Step 3

Simplify the numerator.

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Step 3.1

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

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Step 3.1.1

Factor $4$ out of $24$ .

$\sqrt{\frac{3 \sqrt{4 (6)}}{4}}$

Step 3.1.2

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

$\sqrt{\frac{3 \sqrt{2^{2} \cdot 6}}{4}}$

Step 3.2

Pull terms out from under the radical.

$\sqrt{\frac{3 \cdot 2 \sqrt{6}}{4}}$

Step 3.3

Multiply $3$ by $2$ .

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

Step 4

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

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Step 4.1

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

$\sqrt{\frac{2 (3 \sqrt{6})}{4}}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $2$ out of $4$ .

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 5

Rewrite $\sqrt{\frac{3 \sqrt{6}}{2}}$ as $\frac{\sqrt{3 \sqrt{6}}}{\sqrt{2}}$ .

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

Step 6

Multiply $\frac{\sqrt{3 \sqrt{6}}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{\sqrt{3 \sqrt{6}}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

Add $1$ and $1$ .

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

Step 7.6

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

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Step 7.6.1

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

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

Step 7.6.2

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

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

Step 7.6.3

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

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

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

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

Step 7.6.4.2

Rewrite the expression.

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

Step 7.6.5

Evaluate the exponent.

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

Step 8

Simplify the numerator.

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Step 8.1

Combine using the product rule for radicals.

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

Step 8.2

Multiply $2$ by $3$ .

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.91682931 \dots$