Question

$\frac{10i}{1+2i}$ is equal to

• A. -2i
• B. 2i
• C. $-4+2i$
• D. $4+2i$
• E. 6i

Hint:

Algebra Tip: Never leave $i$ in the denominator! Always clear it out using the complex conjugate, just like rationalizing a square root.

Answer 1

The Correct Option is D

Concept:
To simplify a complex number in a fractional form (division), you must rationalize the denominator. This is done by multiplying both the numerator and the denominator by the complex conjugate of the denominator. The product of a complex number $a+bi$ and its conjugate $a-bi$ results in a real number $a^2+b^2$.

Step 1: Identify the complex fraction and its conjugate.
The given expression is: $$z = \frac{10i}{1+2i}$$ The denominator is $1+2i$. Its complex conjugate is found by changing the sign of the imaginary part, which gives $1-2i$.

Step 2: Multiply numerator and denominator by the conjugate.
$$z = \frac{10i}{1+2i} \times \frac{1-2i}{1-2i}$$

Step 3: Expand the numerator.
Distribute $10i$ across the terms in the numerator's conjugate: $$\text{Numerator} = 10i(1) - 10i(2i)$$ $$\text{Numerator} = 10i - 20i^2$$ Since $i^2 = -1$: $$\text{Numerator} = 10i - 20(-1) = 20 + 10i$$

Step 4: Simplify the denominator.
Use the difference of squares formula $(a+bi)(a-bi) = a^2 + b^2$: $$\text{Denominator} = (1)^2 + (2)^2$$ $$\text{Denominator} = 1 + 4 = 5$$

Step 5: Divide and extract the final standard form.
Combine the simplified numerator and denominator: $$z = \frac{20 + 10i}{5}$$ Divide both the real and imaginary parts by 5: $$z = \frac{20}{5} + \frac{10i}{5} = 4 + 2i$$ Hence the correct answer is (D) $4+2i$.