Does multiplying square roots cancel?

Square Roots: When Does Multiplying Them Really Cancel Things Out?

Square roots. They seem simple enough, right? But trust me, they can be trickier than they look! One spot where people often get snagged is when multiplying them. Does something “cancel” out? Well, let’s get to the bottom of this.

Okay, so there’s this basic rule. Think of it as the golden rule of multiplying square roots: the square root of (a times b) is the same as the square root of a times the square root of b. Got it? In math-speak:

√a * √b = √(a*b)

As long as we’re talking about regular, non-negative numbers, this works like a charm. Basically, you just multiply what’s inside the little root symbol and then, if you can, take the square root of the whole thing.

So, Where Does This “Cancellation” Idea Come From?

Here’s where it gets interesting. Sometimes, when you multiply, you end up with a perfect square inside the root. Like this:

√2 * √8 = √(2*8) = √16 = 4

See what happened? Because 16 is a perfect square (4 times 4), we could simplify it down to just plain old 4. It looks like the square roots vanished, doesn’t it? But really, we just took the square root and got a nice, neat number. No magic, just math!

A Little Trick: Simplify First

Here’s a tip I learned the hard way: Sometimes, it’s easier to simplify each square root before you multiply. Check this out:

√12 * √3 = √(43) * √3 = 2√3 * √3 = 2√(33) = 2√9 = 2*3 = 6

We broke down √12 into 2√3 first. Made the whole thing way easier, right? Especially if you’re dealing with bigger numbers, or you spot those perfect square hiding inside.

Watch Out! Negative Numbers Can Trip You Up

Now, this is super important: This rule goes out the window when you start messing with square roots of negative numbers. Seriously, be careful here! Remember imaginary numbers? That’s where i comes in.

The square root of -1 is i. So, if you’re multiplying square roots of negatives, you have to rewrite them using i first. Like this:

√-a * √-b = (√a * i) * (√b * i) = √(ab) * i² = -√(ab)

That little negative sign at the end is a killer. Forget it, and you’ll get the wrong answer every time. Trust me, I’ve been there! For instance:

√-4 * √-9 = (√4 * i) * (√9 * i) = (2i) * (3i) = 6i² = -6

But if you just blindly multiply:

√-4 * √-9 = √((-4)*(-9)) = √36 = 6

Wrong! See the difference?

The Bottom Line

So, multiplying square roots doesn’t magically “cancel” anything. What really happens is that you might end up with a perfect square that you can simplify. The trick is to know the rules, especially when those sneaky negative numbers are involved. Get this stuff down, and you’ll be a square root ninja in no time!