What is expand (2n+1)^5 ?

The solution to expand (2n+1)^5 is 32n^5+80n^4+80n^3+40n^2+10n+1

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expand (2n+1)^5

expand

(2 n + 1)^{5}

32 n^{5} + 80 n^{4} + 80 n^{3} + 40 n^{2} + 10 n + 1

Solution steps

(2 n + 1)^{5}

Apply binomial theorem:

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

a = 2 n, b = 1

= i = 0 \sum 5 (i 5) (2 n)^{(5 - i)} \cdot 1^{i}

Expand summation

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

Simplify

\frac{5 !}{0 ! ( 5 - 0 ) !} (2 n)^{5} \cdot 1^{0} : 32 n^{5}

Simplify

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

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} (2 n)^{3} \cdot 1^{2} : 80 n^{3}

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} (2 n)^{2} \cdot 1^{3} : 40 n^{2}

Simplify

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

Simplify

\frac{5 !}{5 ! ( 5 - 5 ) !} (2 n)^{0} \cdot 1^{5} : 1

= 32 n^{5} + 80 n^{4} + 80 n^{3} + 40 n^{2} + 10 n + 1

(5 x^{2} y + 2 x)^{3}

7 y^{2} - \frac{11}{3} y - \frac{2}{3}

9 x^{2} - 12 x + 3

68 x^{25} + 68 x^{5}

(8 c - d) (c + 9 d) - 5 (d - 8 c)

What is expand (2n+1)^5 ?
The solution to expand (2n+1)^5 is 32n^5+80n^4+80n^3+40n^2+10n+1