What is expand (2m+n)^5 ?

The solution to expand (2m+n)^5 is 32m^5+80m^4n+80m^3n^2+40m^2n^3+10mn^4+n^5

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expand (2m+n)^5

expand

(2 m + n)^{5}

32 m^{5} + 80 m^{4} n + 80 m^{3} n^{2} + 40 m^{2} n^{3} + 10 m n^{4} + n^{5}

Solution steps

(2 m + n)^{5}

Apply binomial theorem:

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

a = 2 m, b = n

= i = 0 \sum 5 (i 5) (2 m)^{(5 - i)} n^{i}

Expand summation

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

Simplify

\frac{5 !}{0 ! ( 5 - 0 ) !} (2 m)^{5} n^{0} : 32 m^{5}

Simplify

\frac{5 !}{1 ! ( 5 - 1 ) !} (2 m)^{4} n^{1} : 80 m^{4} n

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} (2 m)^{3} n^{2} : 80 m^{3} n^{2}

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} (2 m)^{2} n^{3} : 40 m^{2} n^{3}

Simplify

\frac{5 !}{4 ! ( 5 - 4 ) !} (2 m)^{1} n^{4} : 10 m n^{4}

Simplify

\frac{5 !}{5 ! ( 5 - 5 ) !} (2 m)^{0} n^{5} : n^{5}

= 32 m^{5} + 80 m^{4} n + 80 m^{3} n^{2} + 40 m^{2} n^{3} + 10 m n^{4} + n^{5}

\frac{\frac{17}{18} - \frac{7}{9}}{- \frac{1}{9} - \frac{5}{9}}

(- 8 x^{3} + 6 x^{2} - 3 x - 2) - (10 x^{3} - 12 x^{2} + 2 x - 1)

2 a a^{m} \cdot 3 b a^{1 - m}

a^{2} (b - c) + b^{2} (c - a) + c^{2} (a - b)

4 - (x + 2)^{2}

What is expand (2m+n)^5 ?
The solution to expand (2m+n)^5 is 32m^5+80m^4n+80m^3n^2+40m^2n^3+10mn^4+n^5