What is x^2*| 2/(a^{3/2)}*e^{(-x)/a}|^2 ?

The solution to x^2*| 2/(a^{3/2)}*e^{(-x)/a}|^2 is (4e^{-(2x)/a}x^2)/(a^3)

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simplify x^2| 2/(a^{3/2)}e^{(-x)/a}|^2

simplify

x^{2} \cdot \frac{2}{a ^{\frac{3}{2}}} \cdot e^{\frac{- x}{a}}^{2}

\frac{4 e ^{- \frac{2 x}{a}} x ^{2}}{a ^{3}}

Solution steps

x^{2} \frac{2}{a ^{\frac{3}{2}}} e^{\frac{- x}{a}}^{2}

Apply exponent rule:

a^{m} b^{m} = (ab)^{m},

assuming

a \geq 0, b \geq 0

x^{2} e^{\frac{- x}{a}} \frac{2}{a ^{\frac{3}{2}}}^{2} = (x e^{\frac{- x}{a}} \frac{2}{a ^{\frac{3}{2}}})^{2}

= (x e^{\frac{- x}{a}} \frac{2}{a ^{\frac{3}{2}}})^{2}

Simplify

x \frac{2}{a ^{\frac{3}{2}}} e^{\frac{- x}{a}} : \frac{2 x e ^{- \frac{x}{a}}}{a ^{\frac{3}{2}}}

= \frac{2 x e ^{- \frac{x}{a}}}{a ^{\frac{3}{2}}}^{2}

\frac{2 x e ^{- \frac{x}{a}}}{a ^{\frac{3}{2}}} = \frac{2 e ^{- \frac{x}{a}} x}{a ^{\frac{3}{2}}}

= \frac{2 x e ^{- \frac{x}{a}}}{a ^{\frac{3}{2}}}^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 2 e ^{- \frac{x}{a}} x ) ^{2}}{a ^{\frac{3}{2}} ^{2}}

Apply absolute rule:

∣ x ∣^{2} = x^{2}

, assuming

x \in R

a^{\frac{3}{2}}^{2} = (a^{\frac{3}{2}})^{2}

= \frac{( 2 x e ^{- \frac{x}{a}} ) ^{2}}{( a ^{\frac{3}{2}} ) ^{2}}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

(2 e^{- \frac{x}{a}} x)^{2} = 2^{2} x^{2} (e^{- \frac{x}{a}})^{2}

= \frac{2 ^{2} x ^{2} ( e ^{- \frac{x}{a}} ) ^{2}}{( a ^{\frac{3}{2}} ) ^{2}}

(a^{\frac{3}{2}})^{2} : a^{3}

= \frac{2 ^{2} x ^{2} ( e ^{- \frac{x}{a}} ) ^{2}}{a ^{3}}

(e^{- \frac{x}{a}})^{2} : e^{- \frac{2 x}{a}}

= \frac{2 ^{2} x ^{2} e ^{- \frac{2 x}{a}}}{a ^{3}}

2^{2} = 4

= \frac{4 e ^{- \frac{2 x}{a}} x ^{2}}{a ^{3}}

10 t e^{- 5 t}

e^{8 x}

e^{- s t} sin (- 6 t)

2 \cdot e^{- 2 x}

What is x^2*| 2/(a^{3/2)}*e^{(-x)/a}|^2 ?
The solution to x^2*| 2/(a^{3/2)}*e^{(-x)/a}|^2 is (4e^{-(2x)/a}x^2)/(a^3)