What is expand (v^{1/2}+2)^4 ?

The solution to expand (v^{1/2}+2)^4 is v^2+8v^{3/2}+24v+32v^{1/2}+16

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expand (v^{1/2}+2)^4

expand

(v^{\frac{1}{2}} + 2)^{4}

v^{2} + 8 v^{\frac{3}{2}} + 24 v + 32 v^{\frac{1}{2}} + 16

Solution steps

(v^{\frac{1}{2}} + 2)^{4}

Apply binomial theorem:

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

a = v^{\frac{1}{2}}, b = 2

= i = 0 \sum 4 (i 4) (v^{\frac{1}{2}})^{(4 - i)} \cdot 2^{i}

Expand summation

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

Simplify

\frac{4 !}{0 ! ( 4 - 0 ) !} (v^{\frac{1}{2}})^{4} \cdot 2^{0} : v^{2}

Simplify

\frac{4 !}{1 ! ( 4 - 1 ) !} (v^{\frac{1}{2}})^{3} \cdot 2^{1} : 8 v^{\frac{3}{2}}

Simplify

\frac{4 !}{2 ! ( 4 - 2 ) !} (v^{\frac{1}{2}})^{2} \cdot 2^{2} : 24 v

Simplify

\frac{4 !}{3 ! ( 4 - 3 ) !} (v^{\frac{1}{2}})^{1} \cdot 2^{3} : 32 v^{\frac{1}{2}}

Simplify

\frac{4 !}{4 ! ( 4 - 4 ) !} (v^{\frac{1}{2}})^{0} \cdot 2^{4} : 16

= v^{2} + 8 v^{\frac{3}{2}} + 24 v + 32 v^{\frac{1}{2}} + 16

\frac{x - 4}{( x + 5 ) ( x - 1 )}

d x (6 x + 9) e^{3} x^{6} - 8 x + 5

\frac{3}{( s ^{2} + 4 ) ( s ^{2} + 9 )}

\frac{3}{2} + \frac{1}{3} (x - \frac{5}{4}) - \frac{1}{27} (x - \frac{5}{4})^{2}

- 1 (2 x + y = 7)

What is expand (v^{1/2}+2)^4 ?
The solution to expand (v^{1/2}+2)^4 is v^2+8v^{3/2}+24v+32v^{1/2}+16