Basic Math Examples

$\frac{\sqrt{2 m} \sqrt{5}}{\sqrt{20 m}}$

Step 1

Combine $\sqrt{2 m}$ and $\sqrt{20 m}$ into a single radical.

$\sqrt{\frac{2 m}{20 m}} \sqrt{5}$

Step 2

Reduce the expression $\frac{2 m}{20 m}$ by cancelling the common factors.

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Step 2.1

Factor $2$ out of $2 m$ .

$\sqrt{\frac{2 (m)}{20 m}} \sqrt{5}$

Step 2.2

Factor $2$ out of $20 m$ .

$\sqrt{\frac{2 (m)}{2 (10 m)}} \sqrt{5}$

Step 2.3

Cancel the common factor.

$\sqrt{\frac{2 m}{2 (10 m)}} \sqrt{5}$

Step 2.4

Rewrite the expression.

$\sqrt{\frac{m}{10 m}} \sqrt{5}$

Step 3

Cancel the common factor of $m$ .

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Step 3.1

Cancel the common factor.

$\sqrt{\frac{m}{10 m}} \sqrt{5}$

Step 3.2

Rewrite the expression.

$\sqrt{\frac{1}{10}} \sqrt{5}$

Step 4

Combine using the product rule for radicals.

$\sqrt{\frac{1}{10} \cdot 5}$

Step 5

Combine $\frac{1}{10}$ and $5$ .

$\sqrt{\frac{5}{10}}$

Step 6

Cancel the common factor of $5$ and $10$ .

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Step 6.1

Factor $5$ out of $5$ .

$\sqrt{\frac{5 (1)}{10}}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $5$ out of $10$ .

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

Step 6.2.2

Cancel the common factor.

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

Step 6.2.3

Rewrite the expression.

$\sqrt{\frac{1}{2}}$

Step 7

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$\frac{\sqrt{1}}{\sqrt{2}}$

Step 8

Any root of $1$ is $1$ .

$\frac{1}{\sqrt{2}}$

Step 9

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 10

Combine and simplify the denominator.

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Step 10.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 10.2

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

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

Step 10.3

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

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

Step 10.4

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

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

Step 10.5

Add $1$ and $1$ .

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

Step 10.6

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

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Step 10.6.1

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

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

Step 10.6.2

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

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

Step 10.6.3

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

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

Step 10.6.4

Cancel the common factor of $2$ .

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Step 10.6.4.1

Cancel the common factor.

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

Step 10.6.4.2

Rewrite the expression.

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

Step 10.6.5

Evaluate the exponent.

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

Step 11

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.70710678 \dots$