Basic Math Examples

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

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

Step 1

Rewrite

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

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

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

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

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

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

Step 2

Simplify the numerator.

Tap for more steps...

Step 2.1

Rewrite

24

$24$ as

2^{2} \cdot 6

$2^{2} \cdot 6$ .

Tap for more steps...

Step 2.1.1

Factor

4

$4$ out of

24

$24$ .

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

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

Step 2.1.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

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

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

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

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

Step 2.2

Pull terms out from under the radical.

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

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

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

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

Step 3

Simplify the denominator.

Tap for more steps...

Step 3.1

Rewrite

125

$125$ as

5^{2} \cdot 5

$5^{2} \cdot 5$ .

Tap for more steps...

Step 3.1.1

Factor

25

$25$ out of

125

$125$ .

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

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

Step 3.1.2

Rewrite

25

$25$ as

5^{2}

$5^{2}$ .

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

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

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

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

Step 3.2

Pull terms out from under the radical.

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

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

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

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

Step 4

Multiply

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

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

\frac{\sqrt{5}}{\sqrt{5}}

$\frac{\sqrt{5}}{\sqrt{5}}$ .

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

Tap for more steps...

Step 5.1

Multiply

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

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

\frac{\sqrt{5}}{\sqrt{5}}

$\frac{\sqrt{5}}{\sqrt{5}}$ .

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

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

Step 5.2

Move

\sqrt{5}

$\sqrt{5}$ .

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

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

Step 5.3

Raise

\sqrt{5}

$\sqrt{5}$ to the power of

1

$1$ .

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

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

Step 5.4

Raise

\sqrt{5}

$\sqrt{5}$ to the power of

1

$1$ .

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

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 5.6

Add

1

$1$ and

1

$1$ .

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

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

Step 5.7

Rewrite

{\sqrt{5}}^{2}

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

5

$5$ .

Tap for more steps...

Step 5.7.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{5}

$\sqrt{5}$ as

5^{\frac{1}{2}}

$5^{\frac{1}{2}}$ .

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

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 5.7.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

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

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

Step 5.7.4

Cancel the common factor of

2

$2$ .

Tap for more steps...

Step 5.7.4.1

Cancel the common factor.

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

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

Step 5.7.4.2

Rewrite the expression.

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

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

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

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

Step 5.7.5

Evaluate the exponent.

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

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

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

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

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

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

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

Combine using the product rule for radicals.

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

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

Step 6.2

Multiply

5

$5$ by

6

$6$ .

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

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

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

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

Step 7

Multiply

5

$5$ by

5

$5$ .

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

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

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

Decimal Form:

0.43817804 \dots

$0.43817804 \dots$