How do you use the Binomial theorem to expand $(5+2i)^4$ ?

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Answer 1 · 2018-03-27T14:49:50

$(5+2i)^4=41+840i$

According to Binomial Theorem

$(a+b)^4=C_0^4a^4+C_1^4a^3b+C_2^4a^2b^2+C_3^4ab^3+C_4^4b^4$

= $a^4+4a^3b+6a^2b^2+4ab^3+b^4$

Hence $(5+2i)^4$

= $5^4+4*5^3*2i+6*5^2*(2i)^2+4*5*(2i)^3+(2i)^4$

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

= $625+1000i-600-160i+16$

= $41+840i$

How do you use the Binomial theorem to expand (5+2i)^4(5+2i)^4?