Imaginary numbers

Objective

• Review over roots
• Review over some rules for roots
• Review over how to simplify roots
• Go over what i is
• Do some examples with i
• Homework

So, then what are roots?

A root is our way to undo an exponent.

An exponent tells you how many times you need to multiply a number to get another number.

For example:

Is really just a shorter way of writing:

Now, a root is the opposite of that.

A root tells us how many times a number needs to be multiplied to become the number on the inside.

This sounds really complicated, but let’s use our example.

We know 4 to the 4^th power is 256.

So let’s do the opposite, or take the 4^th root of 256:

What this is asking is:

Which we know is actually:

But, since we only need one of them as an answer, our answer would be:

The rule for even square roots

So, technically what we just got is true.

However, an unwritten rule for even roots is to also include its negative component.

So, for our last example:

Even though we just proved that it is equal to:

It’s also equal to:

But that makes sense right?

If you’re not sure, let’s try it out.

So, as we can see, -4 is also a fourth root of 256.

A FEW RULES TO CONSIDER

Multiplying roots

You can multiply roots together, but only multiply.

An example of this is:

​

We know this works because we can also just solve the root problems and then multiply.

So:

You can also add roots together, however only if they are in the root.

Example:

Which is not the same as:

You can also divide roots together if you need to.

Example:

We know this works because we can also just solve the root problems and then divide.

So:

= 9

= 9

= 10, -10

= 6 + 8

= 14

YOU CAN ALSO BREAK SQUARE ROOTS

SIMPLIFIED PROPERLY

It’s also proper to not leave radicals in the denominator of a fraction.

An example is:

​

(Since having the same number in the numerator and denominator is actually 1)

So we have:

Imaginary numbers

So, we’ve seen some examples, but let’s look at one extra example:

Well, we know how to find this answer, right?

_________

-2 -2

Hmmm, it seems we’ve hit a problem.

What we’re essentially asking is:

what * itself = -25?

Well, we could try -5, but:

And we know that the answer can’t be 5 because:

So then, what times itself can make -25?

Well, let’s look at what -25 actually is:

So, let’s take the square root of this to see what’s up:

Well, we know the square root of 25 is 5, -5.

So:

But does -1 have a square root?

The square root of -1

So, as it turns out, yes.

But, we honestly don’t know what it is, only that it exists, so we deem this number imaginary, and note it as the number i.

So, looking at our last example, we have:

Again, we know the square root of 25 is 5, -5:

And now we know that the square root of -1 is i, so:

Finally, our answer here is:

That’s how we solve it

So that’s it.

We make sure to solve the problem the exact same way as we would if the problem was positive.

And instead, we add the letter i to our answer.

Let’s look at a few more examples.

EXAMPLE 1

EXAMPLE 2

Here’s another example.

Let’s say we have something like:

​

__________

3 3

EXAMPLE 3

Last example.

Let’s say we have something like:

​

___________

5 5