What is expand (a_{1}e+a_{2}e)^3 ?

The solution to expand (a_{1}e+a_{2}e)^3 is e^3a_{1}^3+3e^3a_{1}^2a_{2}+3e^3a_{1}a_{2}^2+e^3a_{2}^3

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expand (a_{1}e+a_{2}e)^3

expand

(a_{1} e + a_{2} e)^{3}

e^{3} a_{1}^{3} + 3 e^{3} a_{1}^{2} a_{2} + 3 e^{3} a_{1} a_{2}^{2} + e^{3} a_{2}^{3}

Solution steps

(a_{1} e + a_{2} e)^{3}

= (e a_{1} + e a_{2})^{3}

Apply Perfect Cube Formula:

(a + b)^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}

a = a_{1} e, b = a_{2} e

= (a_{1} e)^{3} + 3 (a_{1} e)^{2} a_{2} e + 3 a_{1} e (a_{2} e)^{2} + (a_{2} e)^{3}

Simplify

(a_{1} e)^{3} + 3 (a_{1} e)^{2} a_{2} e + 3 a_{1} e (a_{2} e)^{2} + (a_{2} e)^{3} : e^{3} a_{1}^{3} + 3 e^{3} a_{1}^{2} a_{2} + 3 e^{3} a_{1} a_{2}^{2} + e^{3} a_{2}^{3}

= e^{3} a_{1}^{3} + 3 e^{3} a_{1}^{2} a_{2} + 3 e^{3} a_{1} a_{2}^{2} + e^{3} a_{2}^{3}

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What is expand (a_{1}e+a_{2}e)^3 ?
The solution to expand (a_{1}e+a_{2}e)^3 is e^3a_{1}^3+3e^3a_{1}^2a_{2}+3e^3a_{1}a_{2}^2+e^3a_{2}^3