How do you simplify $(a^{3} + 2 a^{2} b - a b^{2} + 3 b^{3}) (4 a b)$ $(a^3+2a^2b-ab^2+3b^3)(4ab)$ ?

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How do you simplify $(a^{3} + 2 a^{2} b - a b^{2} + 3 b^{3}) (4 a b)$ $(a^3+2a^2b-ab^2+3b^3)(4ab)$ ?

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Answer 1 · 2015-08-21T07:25:29

$= 4 a^{4} b + 4 a^{3} b^{2} + 12 a b^{4}$ $=color(blue)(4a^4b + 4a^3b^2 +12ab^4$

Explanation:

$(a^{3} + 2 a^{2} b - a b^{2} + 3 b^{3}) (4 a b)$ $(a^3+2a^2b−ab^2+3b^3)color(blue)((4ab)$

$= (a^{3}) \cdot (4 a b) + (2 a^{2} b) \cdot (4 a b) - (a b^{2}) \cdot (4 a b) + (3 b^{3}) . (4 a b)$ $=(a^3) * color(blue)((4ab)) +(2a^2b) * color(blue)((4ab)) - (ab^2) *color(blue)((4ab)) +(3b^3) .color(blue)((4ab)$

$= 4 a^{3} \cdot a b + 8 a^{2} b \cdot a b - 4 a b^{2} \cdot a b + 12 b^{3} \cdot a b$ $=4a^3*ab + 8a^2b*ab - 4ab^2*ab +12b^3*ab$

By property
$a^{m} \cdot a^{n} = a^{m + n}$ $color(blue)(a^m*a^n=a^(m+n)$
Applying the same to powers of $a$ $a$ and $b$ $b$

$= 4 a^{(3 + 1)} b + 8 a^{(2 + 1)} b^{(1 + 1)} - 4 a^{(1 + 1)} b^{(2 + 1)} + 12 a b^{3 + 1}$ $=4a^((3+1))b + 8a^((2+1))b^((1+1)) - 4a^((1+1))b^((2+1)) +12ab^(3+1)$

$= 4 a^{4} b + 8 a^{3} b^{2} - 4 a^{2} b^{3} + 12 a b^{4}$ $=4a^4b + color(blue)(8a^3b^2 - 4a^2b^3) +12ab^4$

$= 4 a^{4} b + 4 a^{3} b^{2} + 12 a b^{4}$ $=color(blue)(4a^4b + 4a^3b^2 +12ab^4$

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How do you simplify $(a^{3} + 2 a^{2} b - a b^{2} + 3 b^{3}) (4 a b)$ $(a^3+2a^2b-ab^2+3b^3)(4ab)$ ?

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Explanation:

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How do you simplify (a3+2a2b−ab2+3b3)(4ab)(a^3+2a^2b-ab^2+3b^3)(4ab)?

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How do you simplify $(a^{3} + 2 a^{2} b - a b^{2} + 3 b^{3}) (4 a b)$ $(a^3+2a^2b-ab^2+3b^3)(4ab)$ ?