What is expand (5*i+2)^4 ?

The solution to expand (5*i+2)^4 is 41-840i

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expand (5*i+2)^4

expand

(5 \cdot i + 2)^{4}

41 - 840 i

Solution steps

(5 i + 2)^{4}

Apply binomial theorem:

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

a = 5 i, b = 2

= i = 0 \sum 4 (i 4) (5 i)^{(4 - i)} \cdot 2^{i}

Expand summation

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

Simplify

\frac{4 !}{0 ! ( 4 - 0 ) !} (5 i)^{4} \cdot 2^{0} : 625 i^{4}

Simplify

\frac{4 !}{1 ! ( 4 - 1 ) !} (5 i)^{3} \cdot 2^{1} : 1000 i^{3}

Simplify

\frac{4 !}{2 ! ( 4 - 2 ) !} (5 i)^{2} \cdot 2^{2} : 600 i^{2}

Simplify

\frac{4 !}{3 ! ( 4 - 3 ) !} (5 i)^{1} \cdot 2^{3} : 160 i

Simplify

\frac{4 !}{4 ! ( 4 - 4 ) !} (5 i)^{0} \cdot 2^{4} : 16

= 625 i^{4} + 1000 i^{3} + 600 i^{2} + 160 i + 16

Simplify

625 i^{4} + 1000 i^{3} + 600 i^{2} + 160 i + 16 : - 840 i + 41

= - 840 i + 41

Rewrite in standard complex form:

41 - 840 i

= 41 - 840 i

(x^{2} + 21 - 5) (x^{2} - x - 2)

(1 + i)^{21}

(5 x + 4 y = 1, 7) (y - x = 2)

(- t + 4) \cdot (- 3 cos (t))

\frac{a ^{3} - 3 a b ^{2} + 2 b ^{3}}{4 a b ^{3} ( a ^{2} - b ^{2} )}