Basic Math Examples

$\sqrt{\frac{24}{125}}$

Step 1

Rewrite $\sqrt{\frac{24}{125}}$ as $\frac{\sqrt{24}}{\sqrt{125}}$ .

$\frac{\sqrt{24}}{\sqrt{125}}$

Step 2

Simplify the numerator.

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Step 2.1

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

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Step 2.1.1

Factor $4$ out of $24$ .

$\frac{\sqrt{4 (6)}}{\sqrt{125}}$

Step 2.1.2

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

$\frac{\sqrt{2^{2} \cdot 6}}{\sqrt{125}}$

Step 2.2

Pull terms out from under the radical.

$\frac{2 \sqrt{6}}{\sqrt{125}}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $125$ as $5^{2} \cdot 5$ .

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Step 3.1.1

Factor $25$ out of $125$ .

$\frac{2 \sqrt{6}}{\sqrt{25 (5)}}$

Step 3.1.2

Rewrite $25$ as $5^{2}$ .

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

Step 3.2

Pull terms out from under the radical.

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

Step 4

Multiply $\frac{2 \sqrt{6}}{5 \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

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

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{2 \sqrt{6}}{5 \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

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

Step 5.2

Move $\sqrt{5}$ .

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

Step 5.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{2 \sqrt{6} \sqrt{5}}{5 ({\sqrt{5}}^{1} \sqrt{5})}$

Step 5.4

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{2 \sqrt{6} \sqrt{5}}{5 ({\sqrt{5}}^{1} {\sqrt{5}}^{1})}$

Step 5.5

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

$\frac{2 \sqrt{6} \sqrt{5}}{5 {\sqrt{5}}^{1 + 1}}$

Step 5.6

Add $1$ and $1$ .

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

Step 5.7

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 5.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{2 \sqrt{6} \sqrt{5}}{5 {(5^{\frac{1}{2}})}^{2}}$

Step 5.7.2

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

$\frac{2 \sqrt{6} \sqrt{5}}{5 \cdot 5^{\frac{1}{2} \cdot 2}}$

Step 5.7.3

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

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

Step 5.7.4

Cancel the common factor of $2$ .

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Step 5.7.4.1

Cancel the common factor.

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

Step 5.7.4.2

Rewrite the expression.

$\frac{2 \sqrt{6} \sqrt{5}}{5 \cdot 5^{1}}$

Step 5.7.5

Evaluate the exponent.

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

Step 6

Simplify the numerator.

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Step 6.1

Combine using the product rule for radicals.

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

Step 6.2

Multiply $5$ by $6$ .

$\frac{2 \sqrt{30}}{5 \cdot 5}$

Step 7

Multiply $5$ by $5$ .

$\frac{2 \sqrt{30}}{25}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{30}}{25}$

Decimal Form:

$0.43817804 \dots$