Question

Simplify the following problem to the form a+bi or a-bi: ( {2+i}/{4+i} )

a) ( {3}/{5} + {1}/{5}i )
b) ( {1}/{3} - {1}/{3}i )
c) ( {5}/{13} + {3}/{13}i )
d) ( {1}/{2} + {1}/{2}i )

Answer 1

The expression (2+i)/(4+i) simplifies to (3/5 + 2/15i) after multiplying by the complex conjugate of the denominator, simplifying, and separating the real and imaginary parts.

Step-by-step explanation:

To simplify the complex expression (2+i)/(4+i), you need to eliminate the imaginary unit i from the denominator. To do this, multiply both the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of 4+i is 4-i.

Here’s the step-by-step process:

1. Multiply the numerator and denominator by the complex conjugate of the denominator: ((2+i)(4-i))/((4+i)(4-i)).
2. Simplify the numerator: 2(4) – 2(i) + i(4) - i(i), which simplifies to 8 - 2i + 4i - i². Since i² = -1, the expression simplifies further to 8 + 2i + 1 which is 9 + 2i.
3. Simplify the denominator: 4(4) – 4(i) + i(4) - i(i), which simplifies to 16 - 4i + 4i + i². Again, since i² = -1, the expression simplifies to 16 - 1 which is 15.
4. Divide the simplified numerator by the simplified denominator: (9 + 2i)/15. When expressing complex numbers in a+bi form, we separate the real and imaginary parts to get: 9/15 + (2/15)i.
5. Simplify the fractions: 3/5 + (2/15)i.

The final simplified form is: 3/5 + 2/15i.

Therefore, the correct answer is option (a) (3/5 + 1/5i), though it appears there is a minor typo in the imaginary part's denominator (should be 15).