Question

What is the value of 5^3 i^9?

Answer 1

this is what i got

Answer 2

What is 5^3 ?

Answer 3

really lol i have no idea how to do it

Answer 4

(53)(i9)
=125i

Answer 5

125

Answer 6

that is what i got

Answer 7

Have you learned exponents?
What does an exponent of 3 tell you to do?

Answer 8

Correct.
5^3 = 5 * 5 * 5 = 125

Answer 9

Now we need to deal with i^9

Answer 10

im just confused on the i^9

Answer 11

I'll explain i^9 and i to any whole number power.

Any number raised to the 0 power (except 0) equals 1.
That means i^0 = 1

By definition, a number raised to the 1 power equals itself, so
i^1 = i

By definition, i^2 = -1

Ok so far?

Answer 12

yeah

Answer 13

So far we have this:

\(i^0 = 1\)

\(i^1 = i\)

\(i^2 = -1\)

Now we need i^3.

\(i^3 = i^2 \times i = -1 \times i = -i\)

Now we have \(i^3 = -i\)

Now we do one more, \(i^4\)

\(i^4 = i^3 \times i = -i \times i = -i^2 = -(-1) = 1\)

Now let's write the entire thing again, so we can see a pattern:

\(i^0 = 1\)

\(i^1 = i\)

\(i^2 = -1\)

\(i^3 = -i\)

\(i^4 = 1\)

Answer 14

Once we see that \(i^4\) is the same as \(i^0\), we can tell that \(i^5\) will be the same as \(i^1\), etc.
This pattern will continue.

Base Exponent Result
i multiple of 4 1
i (multiple of 4) + 1 i
i (multiple of 4) + 2 -1
i (multiple of 4) + 3 -i

Answer 15

Now we compare 9 to a multiple of 4.
8 is a multiple of 4, since 8 = 2 * 4
9 = 8 + 1
9 is a (multiple of 4) plus 1, in better English, 9 is 1 more than a multiple of 4.

For an exponent of 1 more than a multiple of 4, i raised to that exponent equals i

\(\Large i^9 = i\)

Answer 16

Example:
What is \(\Large i^{253}\) ?

Answer: 253/4 = 63.25, so 253 = 4 * 63 + 1
Since 253 is 1 more than a multiple of 4,

\(\Large i^{253} = i\)