simplify in the form a+bi

Question

5-4i/9+4i

Answer 1

To simplify the expression (5-4i)/(9+4i) in the form a+bi, we need to rationalize the denominator.

To do so, we multiply the numerator and denominator by the conjugate of the denominator, which is 9-4i.

((5-4i)/(9+4i)) * ((9-4i)/(9-4i))

Using the distributive property, we can then simplify:

(45-20i-36i+16i^2) / (81-16i^2)

Simplifying further:

(45-56i+16(-1)) / (81-16(-1))

Now, compute:

(45-56i-16) / (81+16)

Continuing to simplify:

(29-56i) / 97

Therefore, the simplified form of (5-4i)/(9+4i) is:

29/97 - (56/97)i.