What is expand (a-2b)^5 ?

The solution to expand (a-2b)^5 is a^5-10a^4b+40a^3b^2-80a^2b^3+80ab^4-32b^5

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expand (a-2b)^5

expand

(a - 2 b)^{5}

a^{5} - 10 a^{4} b + 40 a^{3} b^{2} - 80 a^{2} b^{3} + 80 a b^{4} - 32 b^{5}

Solution steps

(a - 2 b)^{5}

Apply binomial theorem:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = a, b = - 2 b

= i = 0 \sum 5 (i 5) a^{(5 - i)} (- 2 b)^{i}

Expand summation

= \frac{5 !}{0 ! ( 5 - 0 ) !} a^{5} (- 2 b)^{0} + \frac{5 !}{1 ! ( 5 - 1 ) !} a^{4} (- 2 b)^{1} + \frac{5 !}{2 ! ( 5 - 2 ) !} a^{3} (- 2 b)^{2} + \frac{5 !}{3 ! ( 5 - 3 ) !} a^{2} (- 2 b)^{3} + \frac{5 !}{4 ! ( 5 - 4 ) !} a^{1} (- 2 b)^{4} + \frac{5 !}{5 ! ( 5 - 5 ) !} a^{0} (- 2 b)^{5}

Simplify

\frac{5 !}{0 ! ( 5 - 0 ) !} a^{5} (- 2 b)^{0} : a^{5}

Simplify

\frac{5 !}{1 ! ( 5 - 1 ) !} a^{4} (- 2 b)^{1} : - 10 a^{4} b

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} a^{3} (- 2 b)^{2} : 40 a^{3} b^{2}

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} a^{2} (- 2 b)^{3} : - 80 a^{2} b^{3}

Simplify

\frac{5 !}{4 ! ( 5 - 4 ) !} a^{1} (- 2 b)^{4} : 80 a b^{4}

Simplify

\frac{5 !}{5 ! ( 5 - 5 ) !} a^{0} (- 2 b)^{5} : - 32 b^{5}

= a^{5} - 10 a^{4} b + 40 a^{3} b^{2} - 80 a^{2} b^{3} + 80 a b^{4} - 32 b^{5}

\frac{\frac{x ^{2} + 9 x + 18}{x - 5}}{\frac{x + 3}{x ^{2} - 25}}

\frac{2 + i}{1 + 2 i}

(2 + 3 i) (2 - i)

\frac{1 + 2}{2}

ln (7 + x^{3})

What is expand (a-2b)^5 ?
The solution to expand (a-2b)^5 is a^5-10a^4b+40a^3b^2-80a^2b^3+80ab^4-32b^5