Write The Expression In The Form A+bi

Ever feel like you're juggling imaginary numbers with real numbers, trying to get them to play nice? I know, I know, it sounds like a math circus, but stick with me!

Let's say you've got a math problem that's throwing curveballs like a major league pitcher. Maybe it looks something like this: some numbers mixed with the mysterious "i".

Our mission? To wrangle this mathematical beast and present it in a neat, tidy form: a + bi.

Breaking Down the A+bi Dream Team

Think of a + bi as a perfectly balanced seesaw. On one side, you have "a," the chill, real number. It's the number you know and love, like 5, -2, or even pi (π)!

On the other side, we've got "bi." This is where the fun begins. "i" is our imaginary unit. And "b" is just another real number hanging out with its imaginary friend.

The "i" Factor: Not as Scary as It Sounds

"Imaginary" might sound intimidating, like something straight out of a fantasy novel. But fear not! It's just a label. It represents the square root of -1.

So, if you see 3i, don't picture some ghostly apparition. Just think: "Okay, that's three times the square root of -1."

Operation: A+bi Transformation!

Now, let's say you've got something like (2 + i) + (3 - 2i). Our job is to massage it until it looks like our target form, a + bi.

First, unleash your inner accountant and combine all the real numbers (the "a" parts). 2 + 3? That's a cool 5.

Next, gather up all the imaginary parts (the "bi" bits). i - 2i? That's -1i, or just -i.

Putting it all together? Boom! 5 - i. We did it! That is a perfect a + bi form.

Example: From Mess to Magnificent

Let's tackle another one. Imagine we have (4 - i) * 2. Think of it like distributing pizza slices.

Multiply 2 by 4. That gives us 8. Then, multiply 2 by -i, which is -2i.

Voila! 8 - 2i! Another a + bi success story! I hope you feel like a math wizard now.

Pro Tip: Keep it Simple!

Sometimes, you might have something like i squared (i²). Here is a little secret: i² is equal to -1.

So, if you see i², just swap it out for -1 and keep simplifying! The goal is always to isolate the real and imaginary parts.

Now you're armed and dangerous with the knowledge to conquer complex numbers. Go forth and transform every messy expression into the beautiful form of a + bi!