Basic Math Examples

$\frac{π}{1000 - 2 \sqrt{π}} \cdot \frac{- 1}{\sqrt{π}}$

Step 1

Multiply $\frac{π}{1000 - 2 \sqrt{π}}$ by $\frac{1000 + 2 \sqrt{π}}{1000 + 2 \sqrt{π}}$ .

$\frac{π}{1000 - 2 \sqrt{π}} \cdot \frac{1000 + 2 \sqrt{π}}{1000 + 2 \sqrt{π}} \cdot \frac{- 1}{\sqrt{π}}$

Step 2

Multiply $\frac{π}{1000 - 2 \sqrt{π}}$ by $\frac{1000 + 2 \sqrt{π}}{1000 + 2 \sqrt{π}}$ .

$\frac{π (1000 + 2 \sqrt{π})}{(1000 - 2 \sqrt{π}) (1000 + 2 \sqrt{π})} \cdot \frac{- 1}{\sqrt{π}}$

Step 3

Expand the denominator using the FOIL method.

$\frac{π (1000 + 2 \sqrt{π})}{1000000 + 2000 \sqrt{π} - 2000 \sqrt{π} - 4 {\sqrt{π}}^{2}} \cdot \frac{- 1}{\sqrt{π}}$

Step 4

Simplify.

$\frac{π (1000 + 2 \sqrt{π})}{1000000 - 4 π} \cdot \frac{- 1}{\sqrt{π}}$

Step 5

Cancel the common factor of $1000 + 2 \sqrt{π}$ and $1000000 - 4 π$ .

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Step 5.1

Factor $2$ out of $π (1000 + 2 \sqrt{π})$ .

$\frac{2 (π (500 + \sqrt{π}))}{1000000 - 4 π} \cdot \frac{- 1}{\sqrt{π}}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $2$ out of $1000000$ .

$\frac{2 (π (500 + \sqrt{π}))}{2 (500000) - 4 π} \cdot \frac{- 1}{\sqrt{π}}$

Step 5.2.2

Factor $2$ out of $- 4 π$ .

$\frac{2 (π (500 + \sqrt{π}))}{2 (500000) + 2 (- 2 π)} \cdot \frac{- 1}{\sqrt{π}}$

Step 5.2.3

Factor $2$ out of $2 (500000) + 2 (- 2 π)$ .

$\frac{2 (π (500 + \sqrt{π}))}{2 (500000 - 2 π)} \cdot \frac{- 1}{\sqrt{π}}$

Step 5.2.4

Cancel the common factor.

$\frac{2 (π (500 + \sqrt{π}))}{2 (500000 - 2 π)} \cdot \frac{- 1}{\sqrt{π}}$

Step 5.2.5

Rewrite the expression.

$\frac{π (500 + \sqrt{π})}{500000 - 2 π} \cdot \frac{- 1}{\sqrt{π}}$

Step 6

Move the negative in front of the fraction.

$\frac{π (500 + \sqrt{π})}{500000 - 2 π} \cdot (- \frac{1}{\sqrt{π}})$

Step 7

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

$\frac{π (500 + \sqrt{π})}{500000 - 2 π} \cdot (- (\frac{1}{\sqrt{π}} \cdot \frac{\sqrt{π}}{\sqrt{π}}))$

Step 8

Simplify terms.

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Step 8.1

Combine and simplify the denominator.

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Step 8.1.1

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

$\frac{π (500 + \sqrt{π})}{500000 - 2 π} \cdot (- \frac{\sqrt{π}}{\sqrt{π} \sqrt{π}})$

Step 8.1.2

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

$\frac{π (500 + \sqrt{π})}{500000 - 2 π} \cdot (- \frac{\sqrt{π}}{{\sqrt{π}}^{1} \sqrt{π}})$

Step 8.1.3

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

$\frac{π (500 + \sqrt{π})}{500000 - 2 π} \cdot (- \frac{\sqrt{π}}{{\sqrt{π}}^{1} {\sqrt{π}}^{1}})$

Step 8.1.4

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

$\frac{π (500 + \sqrt{π})}{500000 - 2 π} \cdot (- \frac{\sqrt{π}}{{\sqrt{π}}^{1 + 1}})$

Step 8.1.5

Add $1$ and $1$ .

$\frac{π (500 + \sqrt{π})}{500000 - 2 π} \cdot (- \frac{\sqrt{π}}{{\sqrt{π}}^{2}})$

Step 8.1.6

Rewrite ${\sqrt{π}}^{2}$ as $π$ .

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Step 8.1.6.1

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

$\frac{π (500 + \sqrt{π})}{500000 - 2 π} \cdot (- \frac{\sqrt{π}}{{(π^{\frac{1}{2}})}^{2}})$

Step 8.1.6.2

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

$\frac{π (500 + \sqrt{π})}{500000 - 2 π} \cdot (- \frac{\sqrt{π}}{π^{\frac{1}{2} \cdot 2}})$

Step 8.1.6.3

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

$\frac{π (500 + \sqrt{π})}{500000 - 2 π} \cdot (- \frac{\sqrt{π}}{π^{\frac{2}{2}}})$

Step 8.1.6.4

Cancel the common factor of $2$ .

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Step 8.1.6.4.1

Cancel the common factor.

$\frac{π (500 + \sqrt{π})}{500000 - 2 π} \cdot (- \frac{\sqrt{π}}{π^{\frac{2}{2}}})$

Step 8.1.6.4.2

Rewrite the expression.

$\frac{π (500 + \sqrt{π})}{500000 - 2 π} \cdot (- \frac{\sqrt{π}}{π^{1}})$

Step 8.1.6.5

Simplify.

$\frac{π (500 + \sqrt{π})}{500000 - 2 π} \cdot (- \frac{\sqrt{π}}{π})$

Step 8.2

Cancel the common factor of $π$ .

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Step 8.2.1

Move the leading negative in $- \frac{\sqrt{π}}{π}$ into the numerator.

$\frac{π (500 + \sqrt{π})}{500000 - 2 π} \cdot \frac{- \sqrt{π}}{π}$

Step 8.2.2

Cancel the common factor.

$\frac{π (500 + \sqrt{π})}{500000 - 2 π} \cdot \frac{- \sqrt{π}}{π}$

Step 8.2.3

Rewrite the expression.

$\frac{500 + \sqrt{π}}{500000 - 2 π} \cdot (- \sqrt{π})$

Step 8.3

Combine $\frac{500 + \sqrt{π}}{500000 - 2 π}$ and $\sqrt{π}$ .

$- \frac{(500 + \sqrt{π}) \sqrt{π}}{500000 - 2 π}$

Step 8.4

Apply the distributive property.

$- \frac{500 \sqrt{π} + \sqrt{π} \sqrt{π}}{500000 - 2 π}$

Step 8.5

Combine using the product rule for radicals.

$- \frac{500 \sqrt{π} + \sqrt{π π}}{500000 - 2 π}$

Step 9

Simplify each term.

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Step 9.1

Raise $π$ to the power of $1$ .

$- \frac{500 \sqrt{π} + \sqrt{π^{1} π}}{500000 - 2 π}$

Step 9.2

Raise $π$ to the power of $1$ .

$- \frac{500 \sqrt{π} + \sqrt{π^{1} π^{1}}}{500000 - 2 π}$

Step 9.3

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

$- \frac{500 \sqrt{π} + \sqrt{π^{1 + 1}}}{500000 - 2 π}$

Step 9.4

Add $1$ and $1$ .

$- \frac{500 \sqrt{π} + \sqrt{π^{2}}}{500000 - 2 π}$

Step 9.5

Pull terms out from under the radical, assuming positive real numbers.

$- \frac{500 \sqrt{π} + π}{500000 - 2 π}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \frac{500 \sqrt{π} + π}{500000 - 2 π}$

Decimal Form:

$- 0.00177875 \dots$