Finite Math Examples

$\sqrt{\frac{6 m^{4} n^{5}}{12 m^{5} n^{4}}}$

Step 1

Reduce the expression $\frac{6 m^{4} n^{5}}{12 m^{5} n^{4}}$ by cancelling the common factors.

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Step 1.1

Factor $6$ out of $6 m^{4} n^{5}$ .

$\sqrt{\frac{6 (m^{4} n^{5})}{12 m^{5} n^{4}}}$

Step 1.2

Factor $6$ out of $12 m^{5} n^{4}$ .

$\sqrt{\frac{6 (m^{4} n^{5})}{6 (2 m^{5} n^{4})}}$

Step 1.3

Cancel the common factor.

$\sqrt{\frac{6 (m^{4} n^{5})}{6 (2 m^{5} n^{4})}}$

Step 1.4

Rewrite the expression.

$\sqrt{\frac{m^{4} n^{5}}{2 m^{5} n^{4}}}$

Step 2

Cancel the common factor of $m^{4}$ and $m^{5}$ .

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Step 2.1

Factor $m^{4}$ out of $m^{4} n^{5}$ .

$\sqrt{\frac{m^{4} (n^{5})}{2 m^{5} n^{4}}}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $m^{4}$ out of $2 m^{5} n^{4}$ .

$\sqrt{\frac{m^{4} (n^{5})}{m^{4} (2 m n^{4})}}$

Step 2.2.2

Cancel the common factor.

$\sqrt{\frac{m^{4} n^{5}}{m^{4} (2 m n^{4})}}$

Step 2.2.3

Rewrite the expression.

$\sqrt{\frac{n^{5}}{2 m n^{4}}}$

Step 3

Cancel the common factor of $n^{5}$ and $n^{4}$ .

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Step 3.1

Factor $n^{4}$ out of $n^{5}$ .

$\sqrt{\frac{n^{4} n}{2 m n^{4}}}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $n^{4}$ out of $2 m n^{4}$ .

$\sqrt{\frac{n^{4} n}{n^{4} (2 m)}}$

Step 3.2.2

Cancel the common factor.

$\sqrt{\frac{n^{4} n}{n^{4} (2 m)}}$

Step 3.2.3

Rewrite the expression.

$\sqrt{\frac{n}{2 m}}$

Step 4

Rewrite $\sqrt{\frac{n}{2 m}}$ as $\frac{\sqrt{n}}{\sqrt{2 m}}$ .

$\frac{\sqrt{n}}{\sqrt{2 m}}$

Step 5

Multiply $\frac{\sqrt{n}}{\sqrt{2 m}}$ by $\frac{\sqrt{2 m}}{\sqrt{2 m}}$ .

$\frac{\sqrt{n}}{\sqrt{2 m}} \cdot \frac{\sqrt{2 m}}{\sqrt{2 m}}$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{\sqrt{n}}{\sqrt{2 m}}$ by $\frac{\sqrt{2 m}}{\sqrt{2 m}}$ .

$\frac{\sqrt{n} \sqrt{2 m}}{\sqrt{2 m} \sqrt{2 m}}$

Step 6.2

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

$\frac{\sqrt{n} \sqrt{2 m}}{{\sqrt{2 m}}^{1} \sqrt{2 m}}$

Step 6.3

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

$\frac{\sqrt{n} \sqrt{2 m}}{{\sqrt{2 m}}^{1} {\sqrt{2 m}}^{1}}$

Step 6.4

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

$\frac{\sqrt{n} \sqrt{2 m}}{{\sqrt{2 m}}^{1 + 1}}$

Step 6.5

Add $1$ and $1$ .

$\frac{\sqrt{n} \sqrt{2 m}}{{\sqrt{2 m}}^{2}}$

Step 6.6

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

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Step 6.6.1

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

$\frac{\sqrt{n} \sqrt{2 m}}{{({(2 m)}^{\frac{1}{2}})}^{2}}$

Step 6.6.2

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

$\frac{\sqrt{n} \sqrt{2 m}}{{(2 m)}^{\frac{1}{2} \cdot 2}}$

Step 6.6.3

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

$\frac{\sqrt{n} \sqrt{2 m}}{{(2 m)}^{\frac{2}{2}}}$

Step 6.6.4

Cancel the common factor of $2$ .

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Step 6.6.4.1

Cancel the common factor.

$\frac{\sqrt{n} \sqrt{2 m}}{{(2 m)}^{\frac{2}{2}}}$

Step 6.6.4.2

Rewrite the expression.

$\frac{\sqrt{n} \sqrt{2 m}}{{(2 m)}^{1}}$

Step 6.6.5

Simplify.

$\frac{\sqrt{n} \sqrt{2 m}}{2 m}$

Step 7

Combine using the product rule for radicals.

$\frac{\sqrt{n \cdot 2 m}}{2 m}$

Step 8

Reorder factors in $\frac{\sqrt{n \cdot 2 m}}{2 m}$ .

$\frac{\sqrt{2 n m}}{2 m}$