Simplify the following: a^2 b (a-b^2 )+a b^2 (4 a b-2 a^2 )-a^3 b(1-2 b)

a^2 b (a-b^2 )+a b^2 (4 a b-2 a^2 )-a^3 b(1-2 b) =a^2 b a-a^2 b b^2+a b 2 4 a b-a b 2 2 a b 2-a 3 b 2 b =a^ (2+1) b-a^2 b^ (1+2) +4 a^ (1+1) b^ (2+1) -2 a^ (1+2) b^2-a^3 b+2 a^3 b^ (1+1) =a^3 b-a^2 b^3+4 a^2 b^3-2 a^3 b^2-a^3 b+2 a^3 b^2 =3 a^2 b^3

Simplify the following: a^2 b (a-b^2 )+a b^2 (4 a b-2 a^2 )-a^3 b…

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Question

Simplify the following: $a^2 b\left(a-b^2\right)+a b^2\left(4 a b-2 a^2\right)-a^3 b(1-2 b)$

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Answer

$a^2 b\left(a-b^2\right)+a b^2\left(4 a b-2 a^2\right)-a^3 b(1-2 b)$
$=a^2 b \times a-a^2 b \times b^2+a b 2 \times 4 a b-a b 2 \times 2 a b 2-a 3 b \times 2 b$
$=a^{(2+1)} \times b-a^2 \times b^{(1+2)}+4 \times a^{(1+1)} \times b^{(2+1)}-2 \times a^{(1+2)} \times b^2-a^3 b+2 \times a^3 \times b^{(1+1)}$
$=a^3 b-a^2 b^3+4 a^2 b^3-2 a^3 b^2-a^3 b+2 a^3 b^2$
$=3 a^2 b^3$

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