Simplify the following: .

Break down the radicand so that we can rewrite our root in simple radical form.

Start to simplify the square root by taking the √4, which is, of course, 2.

Finish up by simplifying √-1 = i. Done and done.

By the way, it's a good idea to move that i around to the front of the radical, just so it doesn't look like it's part of the square root. Keep it easy on the eyes, you know?

-2i √2

Simplify the following: i¹⁰.

Rewrite the expression in factors of i².

i¹⁰ = i² i² i² i² i²

Since i² = -1, we can substitute and simplify. There's certainly nothing like multiplying a bunch of -1s together over and over again.

i² i² i² i² i² = (-1)(-1)(-1)(-1)(-1) = -1

Simplify the following: .

Step one is to simplify the numerator. That 49 up top is a perfect square, so things shouldn't be too terrible looking.

That gives us -7i in the numerator. Next, we need to rationalize the denominator. To do so, we need to multiply by √7 over √7. Simplification is the last piece of the puzzle.

Finally, since those -7s cancel out, we can simplify and find our final answer. Again, we'll move the i to the front of the radical sign so we don't get confused.

i √7

Simplify the following: .

First, let's take care of those i 's. To do so, we can just use our good ol' exponent rules. Here we'll need to subtract.

Since every i² = -1, we can rewrite and simplify.

Last but not least, we can simplify our whole numbers. The way you write your final answer is up to you.

or

Simplify the following: i³ + i⁵ + i⁷.

Something is a little different here. We'll start by breaking down the i 's into multiples of i², since i² is STILL equal to -1.

(i × i²) + (i × i²i²) + (i × i²i²i²)

We can change all of the i² factors to -1.

(i × i²) + (i × i²i²) + (i × i²i²i²) = (-1)i + (-1)(-1)i + (-1)(-1)(-1)i

Multiplying out the -1s gives us our next step.

(-1)i + (-1)(-1)i + (-1)(-1)(-1)i = -i + i – i

Finally, we need to add and subtract some i 's. Since –i + i = 0, this is a pretty smooth finish.

-i + i – i = -i