What is expand (4-2ln(t))^4 ?

The solution to expand (4-2ln(t))^4 is 256-512ln(t)+384ln^2(t)-128ln^3(t)+16ln^4(t)

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expand (4-2ln(t))^4

expand

(4 - 2 ln (t))^{4}

256 - 512 ln (t) + 384 ln^{2} (t) - 128 ln^{3} (t) + 16 ln^{4} (t)

Solution steps

(4 - 2 ln (t))^{4}

Apply binomial theorem:

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

a = 4, b = - 2 ln (t)

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

Expand summation

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

\frac{4 !}{0 ! ( 4 - 0 ) !} \cdot 4^{4} (- 2 ln (t))^{0} = 256

Simplify

\frac{4 !}{1 ! ( 4 - 1 ) !} \cdot 4^{3} (- 2 ln (t))^{1} : - 512 ln (t)

Simplify

\frac{4 !}{2 ! ( 4 - 2 ) !} \cdot 4^{2} (- 2 ln (t))^{2} : 384 ln^{2} (t)

Simplify

\frac{4 !}{3 ! ( 4 - 3 ) !} \cdot 4^{1} (- 2 ln (t))^{3} : - 128 ln^{3} (t)

Simplify

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

= 256 - 512 ln (t) + 384 ln^{2} (t) - 128 ln^{3} (t) + 16 ln^{4} (t)

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What is expand (4-2ln(t))^4 ?
The solution to expand (4-2ln(t))^4 is 256-512ln(t)+384ln^2(t)-128ln^3(t)+16ln^4(t)