Algebraic Expression

Question

Which of the following expressions is equivalent to 𝟐+𝟑𝒊/𝟒−𝟐𝒊?

a. 1

10 + 4

5 𝑖

b. 1

10

c. 7

6 + 2

3 𝑖

d. 1

10 + 3

10 𝑖

Answer 1

Rationalizing the denominator using complex conjugates.

Answer 2

1. Let's first get rid of the imaginary numbers in the denominator by multiplying the fraction with a new fraction that has the conjugate of the denominator both at the numerator and the denominator (therefore yielding a multiplication by one that does not change the original fraction's value)

( (2 + 3i)/(4-2i) ) x ( (4+2i)/(4+2i) )

1. The numerator becomes: 8 + 4i + 12i + 6i² = 8 - 6 + 16i = 2 + 16i
2. The denominator becomes: 16 + 8i - 8i - 4i² = 16 + 4 = 20
3. The fraction becomes: (2 + 16i) / 20
4. Split the numerator into two terms by separating the common denominator:

(2/20) + (16i/20)

1. Simplify each term: (1/10) + (4i/5)

The answer is A (it seems the fractions are not visible in the multiple choices but all the terms are in the answer :) )

Answer 3

Multiply numerator and denominator by the conjugate of the denominator:

[(2+3i)(4+2i)]/[(4-2i)(4+2i)]

Using FOIL in the numerator results in:

8 + 4i + 12i + 6i² = 8 + 16i - 6 = 2 + 16i

The product of the binomials in the denominator results in a difference of squares (which turns the denominator into a real number-- the whole point of multiplying by the conjugate):

16 - 4i² = 16 + 4 = 20

So the result is:

(2+16i)/20

Note that every term is divisible by 2:

(1+8i)/10

Or:

1/10 + 4i/5

Or:

0.1 + 0.8i

Here is confirmation on Desmos with Complex Mode turned on:

desmos.com/calculator/ydcskyixj5

Answer 4

Whenever you're given a complex expression with a numerator and a denominator, the first instinct you should have for a simplification method is to multiply by the complex conjugate of the denominator, which is just a fancy way of saying to multiply both the top and bottom of the fraction by the denominator but with the sign flipped.

Complex Conjugate of the Denominator = 4 + 2i

From here, it's a matter of doing the proper FOILs and remembering that i² = -1. You'll find that the numerator simplifies to 2 + 16i and the denominator simplifies to 20.

Separating the fraction into two, you get 2/20 + 16i/20, which simplifies to 1/10 + (4/5)i.

Answer choice a. is the correct one here.

Hope this was helpful.

Answer 5

2+3i/4 -2i

= depends on where you put implicit parentheses

= 2+(3i/4 -2k) = 2- 5i/8 which doesn't match any of the choices

or = (2+3i)/4 - 2i = (2+3i)(4+2i)/(16 +4) = (8 +16i -6)/20 = 2/20 +16i/20 = 1/10 +8i/10 which appears to be Answer A, the 1st option