What is expand (a+b-(at)/(10))^2*t ?

The solution to expand (a+b-(at)/(10))^2*t is 2abt+a^2t-(a^2t^2)/5-(abt^2)/5+(a^2t^3)/(100)+b^2t

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expand (a+b-(at)/(10))^2*t

expand

(a + b - \frac{a t}{10})^{2} \cdot t

2 ab t + a^{2} t - \frac{a ^{2} t ^{2}}{5} - \frac{ab t ^{2}}{5} + \frac{a ^{2} t ^{3}}{100} + b^{2} t

Solution steps

(a + b - \frac{a t}{10})^{2} t

(a + b - \frac{a t}{10})^{2} = 2 ab + a^{2} + \frac{- a ^{2} t - ab t}{5} + \frac{a ^{2} t ^{2}}{100} + b^{2}

= t (a^{2} + 2 ab + \frac{a ^{2} t ^{2}}{100} + \frac{- a ^{2} t - ab t}{5} + b^{2})

= t (2 ab + a^{2} + \frac{- a ^{2} t - ab t}{5} + \frac{a ^{2} t ^{2}}{100} + b^{2})

Distribute parentheses

= t \cdot 2 ab + t a^{2} + t \frac{- a ^{2} t - ab t}{5} + t \frac{a ^{2} t ^{2}}{100} + t b^{2}

= 2 ab t + a^{2} t + \frac{- a ^{2} t - ab t}{5} t + \frac{a ^{2} t ^{2}}{100} t + b^{2} t

Simplify

2 ab t + a^{2} t + \frac{- a ^{2} t - ab t}{5} t + \frac{a ^{2} t ^{2}}{100} t + b^{2} t : 2 ab t + a^{2} t + \frac{- a ^{2} t ^{2} - ab t ^{2}}{5} + \frac{a ^{2} t ^{3}}{100} + b^{2} t

= 2 ab t + a^{2} t + \frac{- a ^{2} t ^{2} - ab t ^{2}}{5} + \frac{a ^{2} t ^{3}}{100} + b^{2} t

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{- a ^{2} t ^{2} - ab t ^{2}}{5} = - \frac{a ^{2} t ^{2}}{5} - \frac{ab t ^{2}}{5}

= 2 ab t + a^{2} t - \frac{a ^{2} t ^{2}}{5} - \frac{ab t ^{2}}{5} + \frac{a ^{2} t ^{3}}{100} + b^{2} t

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What is expand (a+b-(at)/(10))^2*t ?
The solution to expand (a+b-(at)/(10))^2*t is 2abt+a^2t-(a^2t^2)/5-(abt^2)/5+(a^2t^3)/(100)+b^2t