Basic Math Examples

\frac{1}{\sqrt{40}}

$\frac{1}{\sqrt{40}}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite

40

$40$ as

2^{2} \cdot 10

$2^{2} \cdot 10$ .

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Step 1.1.1

Factor

4

$4$ out of

40

$40$ .

\frac{1}{\sqrt{4 (10)}}

$\frac{1}{\sqrt{4 (10)}}$

Step 1.1.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

\frac{1}{\sqrt{2^{2} \cdot 10}}

$\frac{1}{\sqrt{2^{2} \cdot 10}}$

\frac{1}{\sqrt{2^{2} \cdot 10}}

$\frac{1}{\sqrt{2^{2} \cdot 10}}$

Step 1.2

Pull terms out from under the radical.

\frac{1}{2 \sqrt{10}}

$\frac{1}{2 \sqrt{10}}$

\frac{1}{2 \sqrt{10}}

$\frac{1}{2 \sqrt{10}}$

Step 2

Multiply

\frac{1}{2 \sqrt{10}}

$\frac{1}{2 \sqrt{10}}$ by

\frac{\sqrt{10}}{\sqrt{10}}

$\frac{\sqrt{10}}{\sqrt{10}}$ .

\frac{1}{2 \sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}}

$\frac{1}{2 \sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply

\frac{1}{2 \sqrt{10}}

$\frac{1}{2 \sqrt{10}}$ by

\frac{\sqrt{10}}{\sqrt{10}}

$\frac{\sqrt{10}}{\sqrt{10}}$ .

\frac{\sqrt{10}}{2 \sqrt{10} \sqrt{10}}

$\frac{\sqrt{10}}{2 \sqrt{10} \sqrt{10}}$

Step 3.2

Move

\sqrt{10}

$\sqrt{10}$ .

\frac{\sqrt{10}}{2 (\sqrt{10} \sqrt{10})}

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

Step 3.3

Raise

\sqrt{10}

$\sqrt{10}$ to the power of

1

$1$ .

\frac{\sqrt{10}}{2 ({\sqrt{10}}^{1} \sqrt{10})}

$\frac{\sqrt{10}}{2 ({\sqrt{10}}^{1} \sqrt{10})}$

Step 3.4

Raise

\sqrt{10}

$\sqrt{10}$ to the power of

1

$1$ .

\frac{\sqrt{10}}{2 ({\sqrt{10}}^{1} {\sqrt{10}}^{1})}

$\frac{\sqrt{10}}{2 ({\sqrt{10}}^{1} {\sqrt{10}}^{1})}$

Step 3.5

Use the power rule

a^{m} a^{n} = a^{m + n}

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

\frac{\sqrt{10}}{2 {\sqrt{10}}^{1 + 1}}

$\frac{\sqrt{10}}{2 {\sqrt{10}}^{1 + 1}}$

Step 3.6

Add

1

$1$ and

1

$1$ .

\frac{\sqrt{10}}{2 {\sqrt{10}}^{2}}

$\frac{\sqrt{10}}{2 {\sqrt{10}}^{2}}$

Step 3.7

Rewrite

{\sqrt{10}}^{2}

${\sqrt{10}}^{2}$ as

10

$10$ .

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Step 3.7.1

Use

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

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

\sqrt{10}

$\sqrt{10}$ as

10^{\frac{1}{2}}

$10^{\frac{1}{2}}$ .

\frac{\sqrt{10}}{2 {(10^{\frac{1}{2}})}^{2}}

$\frac{\sqrt{10}}{2 {(10^{\frac{1}{2}})}^{2}}$

Step 3.7.2

Apply the power rule and multiply exponents,

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

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

\frac{\sqrt{10}}{2 \cdot 10^{\frac{1}{2} \cdot 2}}

$\frac{\sqrt{10}}{2 \cdot 10^{\frac{1}{2} \cdot 2}}$

Step 3.7.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

\frac{\sqrt{10}}{2 \cdot 10^{\frac{2}{2}}}

$\frac{\sqrt{10}}{2 \cdot 10^{\frac{2}{2}}}$

Step 3.7.4

Cancel the common factor of

2

$2$ .

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Step 3.7.4.1

Cancel the common factor.

$\frac{\sqrt{10}}{2 \cdot 10^{\frac{2}{2}}}$

Step 3.7.4.2

Rewrite the expression.

$\frac{\sqrt{10}}{2 \cdot 10^{1}}$

$\frac{\sqrt{10}}{2 \cdot 10^{1}}$

Step 3.7.5

Evaluate the exponent.

$\frac{\sqrt{10}}{2 \cdot 10}$

$\frac{\sqrt{10}}{2 \cdot 10}$

$\frac{\sqrt{10}}{2 \cdot 10}$

Step 4

Multiply

$2$ by

$10$ .

$\frac{\sqrt{10}}{20}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{10}}{20}$

Decimal Form:

$0.15811388 \dots$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$