6 Simplify the following a 3i 4135 b 2711 3i 2tom141211 23i

6. Simplify the following: a. 3i +4135 b. (2-71)(1 +3i) 2-tom-141-21(1)= 2-3i 1+2i e. (2+30(2-30 (4+ 60)(4-6i) lve the following

Solution

i⁴ⁿ=1 ,i⁴ⁿ⁺¹=i . i⁴ⁿ⁺²=-1,i⁴ⁿ⁺³=-i

i²=-1

(a+bi)*(a-bi)=a²+b²

6)

a)

3i +4i³-5

=3i +4ii²-5

=3i +4i(-1)-5

=3i -4i-5

=-i-5

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b)

(2-7i)(1+3i)

=(2(1+3i))+(-7i(1+3i))

=(2*1)+(2*3i) +(-7i*1)+(-7i*3i)

=2+6i -7i-21i²

=2+6i -7i-(21*(-1))

=2+6i -7i+21

=23-i

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c)

(2-3i)_/(1+2i)

=((2-3i)(1-2i))_{/((1+2i)(1-2i))}

=((2(1-2i))-(3i(1-2i)))_/(1²+2²)

=((2-4i)-(3i-6i²))_/5

=((2-4i)-(3i+6))_/5

=(2-4i-3i-6)_/5

=(-4-7i)_/5

=-(4_/5)-(7_/5)i

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d)

i⁷⁷⁹=i^(4*194)+3

i⁷⁷⁹=-i

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e)

(2+3i)(2-3i)(4+6i)(4-6i)

=(2²+3²)(4²+6²)

=(4+9)(16+36)

=13*52

=676