Question

Write the quotient 13/5-i in the form a+bi.

A 13/5 - 1/5 i
B. 5-13i
C. 5/2 + 1/2 i
D. 65/24 + 13/24 i

Answer 1

To write the quotient 13/5-i in the form a+bi, we multiply by the complex conjugate of the denominator and simplify, leading to the correct form 65/26 + 13/26 i. Option D is correct.

Step-by-step explanation:

The quotient 13/5-i is obtained by rationalizing the denominator. To do this, we multiply both the numerator and the denominator by the complex conjugate of the denominator, which in this case is 5+i. The complex conjugate is used because when we multiply a complex number by its conjugate, the result is a real number.

Here are the steps to find the quotient in the form a+bi:

1. Multiply the numerator and denominator by the complex conjugate of the denominator: (13/(5-i)) × ((5+i)/(5+i))
2. Apply the FOIL method to the numerator and the rule (a-b)(a+b)=a^2-b^2 to the denominator.
3. After simplification, we get 65/26 + 13/26 i as the result.

So, the correct option is D. 65/24 + 13/24 i.