How to Multiply and Divide Complex Numbers
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Multiplying Complex Numbers
Multiply as you would binomials, then replace \(i^2\) with \(-1\).
Dividing Complex Numbers
To divide by \(a+bi\), multiply numerator and denominator by the conjugate \(a-bi\).
Powers of i
The powers repeat every four: \(i,\ -1,\ -i,\ 1\).
Multiplying and Dividing Complex Numbers
Think of this lesson as more than a rule to memorize. Multiplying and Dividing Complex Numbers is about real and imaginary parts working together. A strong student does not rush to the first formula on the page; they pause, identify the structure of the problem, and then choose the tool that matches that structure. That pause is what prevents most mistakes.
Complex numbers use \(i^2=-1\). Treat \(a+bi\) like a two-part number: the real part \(a\) and the imaginary part \(b\).
Here is the teacher way to approach the topic. First, read the problem slowly and underline the information that is actually given. Next, name the target: are you finding a value, simplifying an expression, comparing two quantities, solving for a variable, or interpreting a graph? Once the target is clear, the calculation becomes much less mysterious because every step has a job.
- Read what is given and what is being asked.
- Choose the rule that connects them.
- Substitute carefully and simplify in small steps.
- Check the final answer against the original question.
A helpful habit is to say what each number represents before using it. For example, if a number is a denominator, a radius, a slope, a common difference, or a coefficient, it should not be treated like an ordinary loose number. Its role tells you where it belongs in the formula. This is especially important on ACT-style questions because many wrong answer choices come from using the right numbers in the wrong places.
Another good habit is to keep the work organized vertically. Write one clean line for substitution, one line for simplifying, and one line for the final answer. If the problem has units, keep the units attached. If the problem has signs, exponents, or parentheses, copy them carefully from one line to the next. Most errors in this topic are not caused by a hard idea; they are caused by dropping a negative sign, combining unlike terms, using the wrong denominator, or skipping a check.
When you finish, ask a quick reasonableness question. Should the answer be positive or negative? Should it be larger or smaller than the original number? Does it fit the graph, table, shape, or equation? Can you plug it back into the original problem? This final check turns the lesson from a procedure into understanding.
On a test, the goal is not to write the longest solution. The goal is to write enough clear work that you can see the structure, avoid traps, and recover quickly if one line goes wrong. Practice the examples below with that mindset: identify the type of problem, choose the matching rule, show the substitution, simplify carefully, and check the answer before moving on.
Free printable Worksheets
Exercises for Multiplying and Dividing Complex Numbers
1) \(Simplify (2+i)(3+i).\)
2) \(Simplify (4-i)(2+3i).\)
3) \(Simplify (5+2i)(5-2i).\)
4) \(Simplify (3i)(-4i).\)
5) \(Simplify i^{18}.\)
6) \(Simplify (1+4i)^2.\)
7) \(Simplify (2-3i)^2.\)
8) \(Simplify \frac{6+8i}{2}.\)
9) \(Simplify \frac{1}{i}.\)
10) \(Simplify \frac{3+2i}{1-i}.\)
11) \(Simplify \frac{4-i}{2+i}.\)
12) \(Simplify \frac{5+5i}{1+2i}.\)
13) \(Simplify (2+i)(3-2i)+(1+i).\)
14) \(Simplify i^{47}.\)
15) \(Simplify (3+4i)(-2+i).\)
16) \(Simplify \frac{7-3i}{2-5i}.\)
17) \(Simplify \frac{-2+i}{3+4i}.\)
18) \(Simplify (1-i)^4.\)
19) \(Simplify (-3+2i)^2.\)
20) \(Simplify \frac{(2+i)(1-3i)}{1+i}.\)
1)\(Simplify (2+i)(3+i).\)
Step 1: Multiply or rationalize using the conjugate when needed.
Step 2: Replace \(i^2\) with \(-1\); 6+2i+3i+i^2=5+5i.
Answer: \(5+5i\)
2)\(Simplify (4-i)(2+3i).\)
Step 1: Multiply or rationalize using the conjugate when needed.
Step 2: Replace \(i^2\) with \(-1\); 8+12i-2i-3i^2=11+10i.
Answer: \(11+10i\)
3)\(Simplify (5+2i)(5-2i).\)
Step 1: Multiply or rationalize using the conjugate when needed.
Step 2: Replace \(i^2\) with \(-1\); conjugates multiply to 5^2+2^2.
Answer: \(29\)
4)\(Simplify (3i)(-4i).\)
Step 1: Multiply or rationalize using the conjugate when needed.
Step 2: Replace \(i^2\) with \(-1\); -12i^2=12.
Answer: \(12\)
5)\(Simplify i^{18}.\)
Step 1: Multiply or rationalize using the conjugate when needed.
Step 2: Replace \(i^2\) with \(-1\); 18 has remainder 2 when divided by 4.
Answer: \(-1\)
6)\(Simplify (1+4i)^2.\)
Step 1: Multiply or rationalize using the conjugate when needed.
Step 2: Replace \(i^2\) with \(-1\); 1+8i+16i^2=-15+8i.
Answer: \(-15+8i\)
7)\(Simplify (2-3i)^2.\)
Step 1: Multiply or rationalize using the conjugate when needed.
Step 2: Replace \(i^2\) with \(-1\); 4-12i+9i^2=-5-12i.
Answer: \(-5-12i\)
8)\(Simplify \frac{6+8i}{2}.\)
Step 1: Multiply or rationalize using the conjugate when needed.
Step 2: Replace \(i^2\) with \(-1\); divide each part by 2.
Answer: \(3+4i\)
9)\(Simplify \frac{1}{i}.\)
Step 1: Multiply or rationalize using the conjugate when needed.
Step 2: Replace \(i^2\) with \(-1\); multiply by i/i.
Answer: \(-i\)
10)\(Simplify \frac{3+2i}{1-i}.\)
Step 1: Multiply or rationalize using the conjugate when needed.
Step 2: Replace \(i^2\) with \(-1\); multiply by 1+i.
Answer: \(\frac12+\frac52i\)
11)\(Simplify \frac{4-i}{2+i}.\)
Step 1: Multiply or rationalize using the conjugate when needed.
Step 2: Replace \(i^2\) with \(-1\); multiply by 2-i.
Answer: \(\frac75-\frac65i\)
12)\(Simplify \frac{5+5i}{1+2i}.\)
Step 1: Multiply or rationalize using the conjugate when needed.
Step 2: Replace \(i^2\) with \(-1\); multiply by 1-2i.
Answer: \(3-i\)
13)\(Simplify (2+i)(3-2i)+(1+i).\)
Step 1: Multiply or rationalize using the conjugate when needed.
Step 2: Replace \(i^2\) with \(-1\); the product is 8-i, then add 1+i.
Answer: \(9\)
14)\(Simplify i^{47}.\)
Step 1: Multiply or rationalize using the conjugate when needed.
Step 2: Replace \(i^2\) with \(-1\); 47 has remainder 3 when divided by 4.
Answer: \(-i\)
15)\(Simplify (3+4i)(-2+i).\)
Step 1: Multiply or rationalize using the conjugate when needed.
Step 2: Replace \(i^2\) with \(-1\); -6+3i-8i+4i^2.
Answer: \(-10-5i\)
16)\(Simplify \frac{7-3i}{2-5i}.\)
Step 1: Multiply or rationalize using the conjugate when needed.
Step 2: Replace \(i^2\) with \(-1\); multiply by 2+5i.
Answer: \(1+i\)
17)\(Simplify \frac{-2+i}{3+4i}.\)
Step 1: Multiply or rationalize using the conjugate when needed.
Step 2: Replace \(i^2\) with \(-1\); multiply by 3-4i.
Answer: \(-\frac2{25}+\frac{11}{25}i\)
18)\(Simplify (1-i)^4.\)
Step 1: Multiply or rationalize using the conjugate when needed.
Step 2: Replace \(i^2\) with \(-1\); (1-i)^2=-2i, then square again.
Answer: \(-4\)
19)\(Simplify (-3+2i)^2.\)
Step 1: Multiply or rationalize using the conjugate when needed.
Step 2: Replace \(i^2\) with \(-1\); 9-12i+4i^2.
Answer: \(5-12i\)
20)\(Simplify \frac{(2+i)(1-3i)}{1+i}.\)
Step 1: Multiply or rationalize using the conjugate when needed.
Step 2: Replace \(i^2\) with \(-1\); numerator is 5-5i, then divide by 1+i.
Answer: \(-5i\)
Multiplying and Dividing Complex Numbers Practice Quiz
