Question

`(3 + 4i)/(7 + 2i) = \\`
Simplify. 'i' is an imaginary number.
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Answer 1

`\large\boxed{(29+22i)/(53)=(29)/(53)+(22)/(53)i}`

Explanation:

`(3+4i)/(7+2i)\cdot(7-2i)/(7-2i)\qquad\text{use}\ (a+b)(a-b)=a^2-b^2\\\\=((3+4i)(7-2i))/((7+2i)(7-2i))\qquad\text{use FOIL}\ (a+b)(c+d)=ac+ad+bc+bd\\\\=((3)(7)+(3)(-2i)+(4i)(7)+(4i)(-2i))/(7^2-(2i)^2)\\\\=(21-6i+28i-8i^2)/(49-2^2i^2)\qquad\text{use}\ √(-1)=i\to i^2=-1\\\\=(21-6i+28i-8(-1))/(49-(4)(-1))`

`=(21-6i+28i+8)/(49+4)\qquad\text{combine like terms}\\\\=((21+8)+(-6i+28i))/(53)\\\\=(29+22i)/(53)\\\\=(29)/(53)+(22)/(53)i`