Question

What is the expression in the standard form a + bi when given the complex number 6/(6 - 5i)?

A) 6/11 + 5/11 i
B) 6/11 - 5/11 i
C) 6/11 + 6/11 i
D) 6/11 - 6/11 i

Answer 1

By multiplying the given complex number 6/(6 - 5i) by the complex conjugate of the denominator, we get the standard form a + bi, which is 6/11 + 5/11 i. Hence, the answer is A) 6/11 + 5/11 i.

Step-by-step explanation:

The expression in standard form a + bi for the given complex number 6/(6 - 5i) is achieved by rationalizing the denominator. To do this, we multiply the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of 6 - 5i is 6 + 5i, which when multiplied gives us:

(6 * (6 + 5i)) / ((6 - 5i) * (6 + 5i))

Expanding the denominator using the formula (a - ib)(a + ib) = a² + b², we get:

(36 + 30i) / (36 + 25)

Simplifying further:

(36 + 30i) / 61

Therefore, the expression splits into real and imaginary parts, yielding:

6/11 + 5/11 i.

So, the correct answer is A) 6/11 + 5/11 i.