What is expand (t+1/t)^4 ?

The solution to expand (t+1/t)^4 is t^4+4t^2+6+4/(t^2)+1/(t^4)

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expand (t+1/t)^4

expand

(t + \frac{1}{t})^{4}

t^{4} + 4 t^{2} + 6 + \frac{4}{t ^{2}} + \frac{1}{t ^{4}}

Solution steps

(t + \frac{1}{t})^{4}

Apply binomial theorem:

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

a = t, b = \frac{1}{t}

= i = 0 \sum 4 (i 4) t^{(4 - i)} (\frac{1}{t})^{i}

Expand summation

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

Simplify

\frac{4 !}{0 ! ( 4 - 0 ) !} t^{4} (\frac{1}{t})^{0} : t^{4}

Simplify

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

Simplify

\frac{4 !}{2 ! ( 4 - 2 ) !} t^{2} (\frac{1}{t})^{2} : 6

Simplify

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

Simplify

\frac{4 !}{4 ! ( 4 - 4 ) !} t^{0} (\frac{1}{t})^{4} : \frac{1}{t ^{4}}

= t^{4} + 4 t^{2} + 6 + \frac{4}{t ^{2}} + \frac{1}{t ^{4}}

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5 (5 x^{4} + 8 x^{2} - 7)^{4}

What is expand (t+1/t)^4 ?
The solution to expand (t+1/t)^4 is t^4+4t^2+6+4/(t^2)+1/(t^4)