Hip 2b squared

Yesterday we talked about the order of operations, and today we're going to talk about exponents, roots and maybe even logarithms (gasp!). And more importantly, we'll talk about the laws of exponents and show you why any number raised to the zeroth power is always one. Or is it?

What are exponents? Exponents are simply a shortcut for multiplication. From Wikipedia:

Exponentiation is a mathematical operation, written as bⁿ, involving two numbers, the base b and the exponent n. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, bⁿ is the product of multiplying n bases:

${\displaystyle b^n=\underset{n}{\underbrace{b\times \cdots \times b}}}$

In that case, bⁿ is called the n-th power of b, or b raised to the power n.

The exponent is usually shown as a superscript to the right of the base. Some common exponents have their own names: the exponent 2 (or 2nd power) is called the square of b (b²) or b squared; the exponent 3 (or 3rd power) is called the cube of b (b³) or b cubed. The exponent −1 of b, or 1 / b, is called the reciprocal of b.

When n is a positive integer and b is not zero, b⁻ⁿ is naturally defined as 1/bⁿ, preserving the property bⁿ × b^m = b^n + m.

Since I'm not able to create superscripts I will use the caret (^) to represent exponentiation, such as b^n.

So 2^2 (or two squared...who remembers playing 2 square and 4 square as a kid? How about 9 square?) which is the same as 2 x 2 or 4,  5^3 (five cubed) is 5 x 5 x 5 or 125, 3^9 is 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 (which is a lot!) but according to the calculator in my trusty iPhone 5c the answer is 19,683.

For those of you who hate algebra, you just had a taste of it in the excerpt from Wikipedia: bⁿ
simply means any number can be used as the base such as 2 or 3 or 5 in my examples and any other number can be used as the exponent like 2 or 3 or 9. The letter b makes sense as the base, while n is often used to stand for a number. The superscript e may have made more sense to stand for exponent but the letter e actually stands for a very specific number (2.7182818...). The number e is actually very important and we'll talk about it a little later.

There are certain rules, or laws, that apply to exponents. They'll make sense as we look at them. This table is from Math is Fun.

In the first example, anything to the 1st power is that number. It makes sense if you remember that the exponent tells how many of the base number to write down, putting a multiplication sign in between each one, Since there is only one, there is no multiplication sign.

In the second example, any number raised to the zeroth power (yes, that's a word) is equal to one. We'll show you why in a few minutes.

The 3rd rule is just a special case for the last law before Fractional Exponents, we'll discuss them both then.

Check out this table from Math is Fun:

The first three laws above (x¹ = x, x⁰ = 1 and x^-1 = 1/x) are just part of the natural sequence of exponents. Have a look at this:

Example: Powers of 5
	.. etc..
52	1 × 5 × 5	25
51	1 × 5	5
50	1	1
5-1	1 ÷ 5	0.2
5-2	1 ÷ 5 ÷ 5	0.04
	.. etc..

Look at that table for a while ... notice that positive, zero or negative exponents are really part of the same pattern, i.e. 5 times larger (or 5 times smaller) depending on whether the exponent gets larger (or smaller).

When you multiply 2 numbers with the same base, you simply add the exponents: x^2 * x^3 = x^2+3 or x^5. Let's write it out and see...x * x  (*) x * x * x = x^5. For example: 2^2 * 2^3 = 4 * 8 = 32, which is what 2^5 equals.

When you divide 2 numbers with the same base, you simply subtract the exponents: x^6 / x^2 = x^6-2 or x^4. For example: 2^6 / 2^2 = 64 / 4 = 16, which is what 2^4 equals.

Now we can show how a number raised to the zeroth power equals 1. We all know that any number divided by itself equals 1...such as 4 / 4 = 1. Let's write this as exponents and use the rule to subtract exponents when dividing. 4 / 4 can also be written as 2^2 / 2^2 which is still equal to 1. Now if we subtract the exponents we get 0, so 2^0 = 1, This applies to any number, except maybe zero...zero is kind of strange. According to the rule above, any number raised to the zero power is 1...and some people agree that this is the definition of zero to the zero power. And most software also gives the answer as 1...including my phone.

But we also know that 0 raised to any power is 0, and some people agree with that as the definition of zero to the zero power. Which also makes sense. But if we use the fractional method above, we get 0^0 / 0^0, which is undefined since you can't divide by zero. So, what is the answer? Nobody agrees, although some say it depends on context.

The next rule says that if you raise a power to a power, you multiply the powers together. (x^2)^3 = x^2*3, which equals x^6. Let's use a base of 2...(2^2)^3 = 4^3 which is 64, which is the same as 2^6.

Next is 2 numbers multiplied together and then raised to a power (in algebra, when 2 or more letters are written right next to each other, that means they are multiplied together). (xy)^3 is the same as x^3 * y^3. Let's pretend that x = 2 and y = 3, so (2 * 3)^3 is 6^3 (remember PEMDAS from yesterday) or 216. How about 2^3 * 3^3? We have 8 * 27 or 216!

The next one is the same thing, but with division, so I won't go through it.

The next to last one says that a negative exponent means the reciprocal of the number. x^(-3) is 1/x^3. For example, 2^(-3) = 1 / 2^3 or 1/8. Let's use our division law...we have 1/8, or as exponents we have 2^0 / 2^3 (remember we can only divide if they are the same base and 2^0 is 1). Subtracting 3 from 0, we get -3, so 2^(-3) = 1 / 2^3.

And the very last one shows what we do with fractional exponents. These are a combination of powers and roots. I thought we would talk about roots today, but this article is getting too long. Maybe I'll talk about them next week (I have something else planned for tomorrow)...but this example shows that x^(2/3) is equal to the 3rd root of x and the whole thing is then squared. Let's use a base of 8 this time, so the numbers work out better...8^(2/3) is equal to the 3rd root of 8, which is 2, square that which is 4. Next week, I'll show why fractional exponents means to take the root of a number (and I'll tell what roots are).

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