What is expand (R+t/2)^4 ?

The solution to expand (R+t/2)^4 is (16R^4+t^4}{16}+2R^3t+\frac{3R^2t^2+Rt^3)/2

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expand (R+t/2)^4

expand

(R + \frac{t}{2})^{4}

\frac{16 R ^{4} + t ^{4}}{16} + 2 R^{3} t + \frac{3 R ^{2} t ^{2} + R t ^{3}}{2}

Solution steps

(R + \frac{t}{2})^{4}

Apply binomial theorem:

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

a = R, b = \frac{t}{2}

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

Expand summation

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

= R^{4} + 2 R^{3} t + \frac{3 R ^{2} t ^{2}}{2} + \frac{R t ^{3}}{2} + \frac{t ^{4}}{16}

Simplify

R^{4} + 2 R^{3} t + \frac{3 R ^{2} t ^{2}}{2} + \frac{R t ^{3}}{2} + \frac{t ^{4}}{16} : \frac{16 R ^{4} + t ^{4}}{16} + 2 R^{3} t + \frac{3 R ^{2} t ^{2} + R t ^{3}}{2}

= \frac{16 R ^{4} + t ^{4}}{16} + 2 R^{3} t + \frac{3 R ^{2} t ^{2} + R t ^{3}}{2}

\frac{a + 3}{( a + 8 ) ( a - 3 )}

(2 + 3)^{10} (2 - 3)^{10}

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

(2 - π)^{2}

z^{2} (\frac{z}{z - 3} - 1 - \frac{3}{z})

What is expand (R+t/2)^4 ?
The solution to expand (R+t/2)^4 is (16R^4+t^4}{16}+2R^3t+\frac{3R^2t^2+Rt^3)/2