What is expand (x+4z)^5 ?

The solution to expand (x+4z)^5 is x^5+20x^4z+160x^3z^2+640x^2z^3+1280xz^4+1024z^5

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expand (x+4z)^5

expand

(x + 4 z)^{5}

x^{5} + 20 x^{4} z + 160 x^{3} z^{2} + 640 x^{2} z^{3} + 1280 x z^{4} + 1024 z^{5}

Solution steps

(x + 4 z)^{5}

Apply binomial theorem:

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

a = x, b = 4 z

= i = 0 \sum 5 (i 5) x^{(5 - i)} (4 z)^{i}

Expand summation

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

Simplify

\frac{5 !}{0 ! ( 5 - 0 ) !} x^{5} (4 z)^{0} : x^{5}

Simplify

\frac{5 !}{1 ! ( 5 - 1 ) !} x^{4} (4 z)^{1} : 20 x^{4} z

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} x^{3} (4 z)^{2} : 160 x^{3} z^{2}

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} x^{2} (4 z)^{3} : 640 x^{2} z^{3}

Simplify

\frac{5 !}{4 ! ( 5 - 4 ) !} x^{1} (4 z)^{4} : 1280 x z^{4}

Simplify

\frac{5 !}{5 ! ( 5 - 5 ) !} x^{0} (4 z)^{5} : 1024 z^{5}

= x^{5} + 20 x^{4} z + 160 x^{3} z^{2} + 640 x^{2} z^{3} + 1280 x z^{4} + 1024 z^{5}

16 x^{8} + 8 x^{4} + 1

x^{3} - 28 x - 48

- x^{3} + 3 x^{2} + 13 x - 15

\frac{x}{2} - \frac{2}{x}

3 a^{16}

What is expand (x+4z)^5 ?
The solution to expand (x+4z)^5 is x^5+20x^4z+160x^3z^2+640x^2z^3+1280xz^4+1024z^5