What is expand (x+4n)^4 ?

The solution to expand (x+4n)^4 is x^4+16nx^3+96n^2x^2+256n^3x+256n^4

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expand (x+4n)^4

expand

(x + 4 n)^{4}

x^{4} + 16 n x^{3} + 96 n^{2} x^{2} + 256 n^{3} x + 256 n^{4}

Solution steps

(x + 4 n)^{4}

Apply binomial theorem:

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

a = x, b = 4 n

= i = 0 \sum 4 (i 4) x^{(4 - i)} (4 n)^{i}

Expand summation

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

Simplify

\frac{4 !}{0 ! ( 4 - 0 ) !} x^{4} (4 n)^{0} : x^{4}

Simplify

\frac{4 !}{1 ! ( 4 - 1 ) !} x^{3} (4 n)^{1} : 16 n x^{3}

Simplify

\frac{4 !}{2 ! ( 4 - 2 ) !} x^{2} (4 n)^{2} : 96 n^{2} x^{2}

Simplify

\frac{4 !}{3 ! ( 4 - 3 ) !} x^{1} (4 n)^{3} : 256 n^{3} x

Simplify

\frac{4 !}{4 ! ( 4 - 4 ) !} x^{0} (4 n)^{4} : 256 n^{4}

= x^{4} + 16 n x^{3} + 96 n^{2} x^{2} + 256 n^{3} x + 256 n^{4}

1 (\frac{3}{2} + \frac{23}{2} \cdot 3 \cdot \frac{3}{2}) + (\frac{3}{2} + \frac{23}{2} \cdot a \cdot 1 \cdot \frac{3}{2})

64 + a^{12}

k^{5} - 7 k^{3} - 5 k^{2} + 35

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

7 a^{3} + 56 b^{3}

What is expand (x+4n)^4 ?
The solution to expand (x+4n)^4 is x^4+16nx^3+96n^2x^2+256n^3x+256n^4