What is expand (1+2e^t)^3 ?

The solution to expand (1+2e^t)^3 is 1+6e^t+12e^{2t}+8e^{3t}

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expand (1+2e^t)^3

expand

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

1 + 6 e^{t} + 12 e^{2 t} + 8 e^{3 t}

Solution steps

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

Apply Perfect Cube Formula:

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

a = 1, b = 2 e^{t}

= 1^{3} + 3 \cdot 1^{2} \cdot 2 e^{t} + 3 \cdot 1 \cdot (2 e^{t})^{2} + (2 e^{t})^{3}

Simplify

1^{3} + 3 \cdot 1^{2} \cdot 2 e^{t} + 3 \cdot 1 \cdot (2 e^{t})^{2} + (2 e^{t})^{3} : 1 + 6 e^{t} + 12 e^{2 t} + 8 e^{3 t}

= 1 + 6 e^{t} + 12 e^{2 t} + 8 e^{3 t}

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What is expand (1+2e^t)^3 ?
The solution to expand (1+2e^t)^3 is 1+6e^t+12e^{2t}+8e^{3t}