What is expand (-3+sqrt(5))^4 ?

The solution to expand (-3+sqrt(5))^4 is 376-168sqrt(5)

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expand (-3+sqrt(5))^4

expand

(- 3 + 5)^{4}

376 - 1685

Solution steps

(- 3 + 5)^{4}

Apply binomial theorem:

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

a = - 3, b = 5

= i = 0 \sum 4 (i 4) (- 3)^{(4 - i)} (5)^{i}

Expand summation

= \frac{4 !}{0 ! ( 4 - 0 ) !} (- 3)^{4} (5)^{0} + \frac{4 !}{1 ! ( 4 - 1 ) !} (- 3)^{3} (5)^{1} + \frac{4 !}{2 ! ( 4 - 2 ) !} (- 3)^{2} (5)^{2} + \frac{4 !}{3 ! ( 4 - 3 ) !} (- 3)^{1} (5)^{3} + \frac{4 !}{4 ! ( 4 - 4 ) !} (- 3)^{0} (5)^{4}

Simplify

\frac{4 !}{0 ! ( 4 - 0 ) !} (- 3)^{4} (5)^{0} : 81

Simplify

\frac{4 !}{1 ! ( 4 - 1 ) !} (- 3)^{3} (5)^{1} : - 1085

Simplify

\frac{4 !}{2 ! ( 4 - 2 ) !} (- 3)^{2} (5)^{2} : 270

Simplify

\frac{4 !}{3 ! ( 4 - 3 ) !} (- 3)^{1} (5)^{3} : - 605

Simplify

\frac{4 !}{4 ! ( 4 - 4 ) !} (- 3)^{0} (5)^{4} : 25

= 81 - 1085 + 270 - 605 + 25

81 - 1085 + 270 - 605 + 25 = 376 - 1685

= 376 - 1685

\frac{( x - 2 ) ( x - 3 )}{6}

(x + 1) (x^{2} + \frac{1}{5})

\frac{3}{4} - \frac{2}{5} + \frac{3}{7}

(e^{- x} + e^{2 x})^{2}

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

What is expand (-3+sqrt(5))^4 ?
The solution to expand (-3+sqrt(5))^4 is 376-168sqrt(5)