Evaluate 80000/((4.5)^(3/2))e^(-(30/4.5+4.5-2))

$\frac{80000}{{(4.5)}^{\frac{3}{2}}} e^{- (\frac{30}{4.5} + 4.5 - 2)}$

Step 1

Simplify the denominator.

Rewrite $4.5$ as ${2.12132034}^{2}$ .

$\frac{80000}{{({2.12132034}^{2})}^{\frac{3}{2}}} \cdot e^{- (\frac{30}{4.5} + 4.5 - 2)}$

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

$\frac{80000}{{2.12132034}^{2 (\frac{3}{2})}} \cdot e^{- (\frac{30}{4.5} + 4.5 - 2)}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{80000}{{2.12132034}^{2 (\frac{3}{2})}} \cdot e^{- (\frac{30}{4.5} + 4.5 - 2)}$

Rewrite the expression.

$\frac{80000}{{2.12132034}^{3}} \cdot e^{- (\frac{30}{4.5} + 4.5 - 2)}$

Raise $2.12132034$ to the power of $3$ .

$\frac{80000}{9.54594154} \cdot e^{- (\frac{30}{4.5} + 4.5 - 2)}$

Step 2

Divide $80000$ by $9.54594154$ .

$8380.52481406 \cdot e^{- (\frac{30}{4.5} + 4.5 - 2)}$

Step 3

Find the common denominator.

Multiply $\frac{30}{4.5}$ by $\frac{\frac{9}{4.5}}{\frac{9}{4.5}}$ .

$8380.52481406 \cdot e^{- (\frac{30}{4.5} \cdot \frac{\frac{9}{4.5}}{\frac{9}{4.5}} + 4.5 - 2)}$

Multiply $\frac{30}{4.5}$ by $\frac{\frac{9}{4.5}}{\frac{9}{4.5}}$ .

$8380.52481406 \cdot e^{- (\frac{30 (\frac{9}{4.5})}{4.5 (\frac{9}{4.5})} + 4.5 - 2)}$

Write $4.5$ as a fraction with denominator $1$ .

$8380.52481406 \cdot e^{- (\frac{30 (\frac{9}{4.5})}{4.5 (\frac{9}{4.5})} + \frac{4.5}{1} - 2)}$

Multiply $\frac{4.5}{1}$ by $\frac{9}{9}$ .

$8380.52481406 \cdot e^{- (\frac{30 (\frac{9}{4.5})}{4.5 (\frac{9}{4.5})} + \frac{4.5}{1} \cdot \frac{9}{9} - 2)}$

Multiply $\frac{4.5}{1}$ by $\frac{9}{9}$ .

$8380.52481406 \cdot e^{- (\frac{30 (\frac{9}{4.5})}{4.5 (\frac{9}{4.5})} + \frac{4.5 \cdot 9}{9} - 2)}$

Write $- 2$ as a fraction with denominator $1$ .

$8380.52481406 \cdot e^{- (\frac{30 (\frac{9}{4.5})}{4.5 (\frac{9}{4.5})} + \frac{4.5 \cdot 9}{9} + \frac{- 2}{1})}$

Multiply $\frac{- 2}{1}$ by $\frac{9}{9}$ .

$8380.52481406 \cdot e^{- (\frac{30 (\frac{9}{4.5})}{4.5 (\frac{9}{4.5})} + \frac{4.5 \cdot 9}{9} + \frac{- 2}{1} \cdot \frac{9}{9})}$

Multiply $\frac{- 2}{1}$ by $\frac{9}{9}$ .

$8380.52481406 \cdot e^{- (\frac{30 (\frac{9}{4.5})}{4.5 (\frac{9}{4.5})} + \frac{4.5 \cdot 9}{9} + \frac{- 2 \cdot 9}{9})}$

Combine $4.5$ and $\frac{9}{4.5}$ .

$8380.52481406 \cdot e^{- (\frac{30 (\frac{9}{4.5})}{\frac{4.5 \cdot 9}{4.5}} + \frac{4.5 \cdot 9}{9} + \frac{- 2 \cdot 9}{9})}$

Multiply $4.5$ by $9$ .

$8380.52481406 \cdot e^{- (\frac{30 (\frac{9}{4.5})}{\frac{40.5}{4.5}} + \frac{4.5 \cdot 9}{9} + \frac{- 2 \cdot 9}{9})}$

Step 4

Combine the numerators over the common denominator.

$8380.52481406 \cdot e^{- (\frac{30 (\frac{9}{4.5})}{\frac{40.5}{4.5}} + \frac{4.5 \cdot 9 - 2 \cdot 9}{9})}$

Step 5

Simplify each term.

Multiply $4.5$ by $9$ .

$8380.52481406 \cdot e^{- (\frac{30 (\frac{9}{4.5})}{\frac{40.5}{4.5}} + \frac{40.5 - 2 \cdot 9}{9})}$

Multiply $- 2$ by $9$ .

$8380.52481406 \cdot e^{- (\frac{30 (\frac{9}{4.5})}{\frac{40.5}{4.5}} + \frac{40.5 - 18}{9})}$

Step 6

Subtract $18$ from $40.5$ .

$8380.52481406 \cdot e^{- (\frac{30 (\frac{9}{4.5})}{\frac{40.5}{4.5}} + \frac{22.5}{9})}$

Step 7

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

$8380.52481406 \cdot e^{- (30 (\frac{9}{4.5}) \frac{4.5}{40.5} + \frac{22.5}{9})}$

Divide $9$ by $4.5$ .

$8380.52481406 \cdot e^{- (30 \cdot 2 \frac{4.5}{40.5} + \frac{22.5}{9})}$

Multiply $30$ by $2$ .

$8380.52481406 \cdot e^{- (60 (\frac{4.5}{40.5}) + \frac{22.5}{9})}$

Divide $4.5$ by $40.5$ .

$8380.52481406 \cdot e^{- (60 \cdot 0.\overline{1} + \frac{22.5}{9})}$

Multiply $60$ by $0.\overline{1}$ .

$8380.52481406 \cdot e^{- (6.\overline{6} + \frac{22.5}{9})}$

Divide $22.5$ by $9$ .

$8380.52481406 \cdot e^{- (6.\overline{6} + 2.5)}$

Step 8

Simplify the expression.

Add $6.\overline{6}$ and $2.5$ .

$8380.52481406 \cdot e^{- 1 \cdot 9.1\overline{6}}$

Multiply $- 1$ by $9.1\overline{6}$ .

$8380.52481406 \cdot e^{- 9.1\overline{6}}$

Step 9

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$8380.52481406 \cdot \frac{1}{e^{9.1\overline{6}}}$

Step 10

Combine $8380.52481406$ and $\frac{1}{e^{9.1\overline{6}}}$ .

$\frac{8380.52481406}{e^{9.1\overline{6}}}$

Step 11

Replace $e$ with an approximation.

$\frac{8380.52481406}{{2.71828182}^{9.1\overline{6}}}$

Step 12

Raise $2.71828182$ to the power of $9.1\overline{6}$ .

$\frac{8380.52481406}{9572.66257416}$

Step 13

Divide $8380.52481406$ by $9572.66257416$ .

$0.87546434$

Evaluate 80000/((4.5)^(3/2))e^(-(30/4.5+4.5-2))

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