What is expand (sqrt(t)+2t)^4 ?

The solution to expand (sqrt(t)+2t)^4 is t^2+8sqrt(t^5)+24t^3+32sqrt(t^7)+16t^4

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expand (sqrt(t)+2t)^4

expand

(t + 2 t)^{4}

t^{2} + 8 t^{5} + 24 t^{3} + 32 t^{7} + 16 t^{4}

Solution steps

(t + 2 t)^{4}

Apply binomial theorem:

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

a = t, b = 2 t

= i = 0 \sum 4 (i 4) (t)^{(4 - i)} (2 t)^{i}

Expand summation

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

Simplify

\frac{4 !}{0 ! ( 4 - 0 ) !} (t)^{4} (2 t)^{0} : t^{2}

Simplify

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

Simplify

\frac{4 !}{2 ! ( 4 - 2 ) !} (t)^{2} (2 t)^{2} : 24 t^{3}

Simplify

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

Simplify

\frac{4 !}{4 ! ( 4 - 4 ) !} (t)^{0} (2 t)^{4} : 16 t^{4}

= t^{2} + 8 t (t)^{3} + 24 t^{3} + 32 t^{3} t + 16 t^{4}

8 t (t)^{3} = 8 t^{\frac{5}{2}}

32 t^{3} t = 32 t^{\frac{7}{2}}

= t^{2} + 8 t^{\frac{5}{2}} + 24 t^{3} + 32 t^{\frac{7}{2}} + 16 t^{4}

Simplify

t^{2} + 8 t^{\frac{5}{2}} + 24 t^{3} + 32 t^{\frac{7}{2}} + 16 t^{4} : t^{2} + 8 t^{5} + 24 t^{3} + 32 t^{7} + 16 t^{4}

= t^{2} + 8 t^{5} + 24 t^{3} + 32 t^{7} + 16 t^{4}

3 y^{2} (6 - 3 y) - \frac{1}{6} y (6 - 3 y)^{3} - \frac{3}{2} y^{3} (6 - 3 y)

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

343 \cdot 63

(\frac{2 m}{3} + 7 b) (\frac{2 m}{3} - 7 b)

(x + \frac{1}{2}) (x + \frac{11}{2})

What is expand (sqrt(t)+2t)^4 ?
The solution to expand (sqrt(t)+2t)^4 is t^2+8sqrt(t^5)+24t^3+32sqrt(t^7)+16t^4