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

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

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Answer 1 · 2018-03-27T14:49:50

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

Explanation:

According to Binomial Theorem

${(a + b)}^{4} = C_{0}^{4} a^{4} + C_{1}^{4} a^{3} b + C_{2}^{4} a^{2} b^{2} + C_{3}^{4} a b^{3} + C_{4}^{4} b^{4}$ $(a+b)^4=C_0^4a^4+C_1^4a^3b+C_2^4a^2b^2+C_3^4ab^3+C_4^4b^4$

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

Hence ${(5 + 2 i)}^{4}$ $(5+2i)^4$

= $5^{4} + 4 \cdot 5^{3} \cdot 2 i + 6 \cdot 5^{2} \cdot {(2 i)}^{2} + 4 \cdot 5 \cdot {(2 i)}^{3} + {(2 i)}^{4}$ $5^4+4*5^3*2i+6*5^2*(2i)^2+4*5*(2i)^3+(2i)^4$

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

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

= $41 + 840 i$ $41+840i$

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

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Explanation:

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