Powers and exponents

I. Exponent expressions
   A. Integer exponents
   B. Fractional exponents
II. Power functions
    A. General appearance
    B. Solving power equations

I. Exponent expressions

A. Integer exponents

We've often seen expressions that look like x^2 or x^3: x raised to
some power. In this class we'll learn some rules for manipulating
expressions like this. First just some terminology: the thing being
raised to a power is called the base, and the power is called the
exponent.

Recall that a power means to multiply something by itself, e.g. 6^3
means 6 . 6 . 6. This immediately leads to rules for how to simplify
expressions in which exponentials with the same base are multiplied by
one another.

6^3 . 6^4 = (6 . 6 . 6) . (6 . 6 . 6 . 6) = 6^7

It's just a matter of counting. The same thing applies to x's or any
other letter:

x^2 . x ^3 = (x . x) . (x . x . x) = x^5

The general rule is

x^a . x^b = x^(a+b)

Just remember that, when you multiply powers, the exponents add. This
only works if the bases are the same, though.

Using similar arguments we get rules for dividing powers:

6^4 / 6^2 = (6 . 6 . 6 . 6) / (6 . 6) = 6^2

In this cases the things on the top and bottom cancel, and the rule is
to subtract instead of add.

x^a / x^b = x^(a-b)

Notice that we'd like this rule to work with negative powers too, so
we define negative powers appropriately:

6^3 / 6^4 = (6 . 6 . 6) / (6 . 6 . 6 . 6) = 1/6 = 6^(-1)
6^3 / 6^5 = (6 . 6 . 6) / (6 . 6 . 6 . 6 . 6) = 1/(6 . 6) = 1/6^2 = 6^(-2)

Thus something to a negative power is just one over that thing to a
positive power.

Finally, we can write rules for raising a power to a power:

(6^2)^3 = (6 . 6)^3 = (6 . 6) . (6 . 6) . (6 . 6) = 6^6

The rule here is that when you raise a power to a power, you multiply
the exponents:

(x^a)^b = x^(a.b)


B. Fractional exponents

The rules for raising powers to powers suggest a natural
generalization: fractional powers. A fractional power is just another
way of writing a root.

For example,

sqrt(4) = 2
sqrt(4)^2 = 4

This is therefore natural to define

sqrt(4) = 4^(1/2)

because then

[4^(1/2)]^2 = 4^[(1/2).2] = 4^1 = 4

That's all there is to a power that looks like 1/n: it just means take
the nth root. Thus sqrt(x) = x^(1/2), cbrt(x) = x^(1/3), and so
forth. All the exponent rules continue to apply.

We can also use these rules to understand things that look like x^(a/b).
This just means (x^a)^(1/b), since a/b = a . (1/b). Thus for example

8^(2/3) = (8^2)^(1/3) = 64^(1/3) = 4

An important note is that you cannot simply add fractional powers:

sqrt(10) + sqrt(10) != sqrt(20)

Instead, you can combine them like you do variables:

sqrt(10) + sqrt(10) = 2 sqrt(10)

The same is true for fractional powers:

x^(1/2) + x^(1/2) = 2 x^(1/2)

You can, however, multiply equal roots by adding the fractional
exponents:

sqrt(x) . sqrt(x) = x^(1/2) . x^(1/2) = x^1 = x


II. Power functions

A. General appearance

Now that we know something about exponential expressions, let's
consider functions involving them. An power function is something that
looks like

f(x) = k x^p

where k and p are numbers. We call k the coefficient and p the
exponent.

What do functions of this sort look like? Well, let's try graphing an
example:

(graph f(x) = x^2)

Clearly this is not a straight line, and it instead curves.

The bigger the exponent, the sharper the curve:

(graph f(x) = x^3, x^4)

Also, notice something else: the x^2 and x^4 cases were always
positive. This isn't surprising, since x^2 is always positive: it's a
number times itself, and a positive times a positive or a negative
times a negative is always positive.

On the other hand, x^3 is negative for negative x. This is a general
rule: positive power functions are always positive (or negative if k
is negative), while odd power functions are both positive and
negative.

Notice that the power p = 1 is a special case, since it's just a
straight line, and it obeys this rule: lines are both positive and
negative.

The effect of the coefficient is to scale the function up or down,
i.e. squash it or stretch it, while leaving the overall shape the
same.

(graph f(x) = (1/2) x^2, x^2, and 2x^2)

We can also consider negative powers:

(graph f(x) = x^-1 and x^-2)

These are like positive powers, except that the curve the opposite
direction, so that they get smaller as x gets larger.

Finally, we can consider fractional powers. As one might expect, these
are just intermediate between integer powers, coming out with a
level of curvature in between the adjacent integers.

(graph f(x) = x^(1/2), x^(3/2))


B. Solving power equations

As we did with lines, it is often useful to solve equations involving
powers. As an example, suppose we have

x^2 = 9

We can use our knowledge of power rules to solve this by taking a
square root, or equivalently by raising to a power of 1/2:

(x^2)^(1/2) = 9^(1/2)
x^1 = 3

We might think we're done here, but there's a trick. Remember what a
graph of x^2 looks like, and remember that there are *two* x values
that produce 9: -3 and +3. This means that -3 is an equally good
solution.

This property of even powers leads to a rule: when solving equations
involving even integer powers, you must always break them up into
positive and negative parts after taking the root.

Thus

x^4 = 16 ==>
x = +16^(1/4) or x = -16^(1/4)
x = +2 or x = -2

This doesn't apply to odd integer powers:

x^3 = 27
x = 27^(1/3) = 3

Notice that -3 isn't a valid solution here, because (-3)^3 = -27, not
+27.

Equations involving fractional powers behave much the same, and one
undoes the fractional power by raising to the inverse power. For
example:

x^(1/2) = 3
[x^(1/2)]^2 = 3^2
x = 9

Similarly

x^(2/3) = 4
[x^(2/3)]^(3/2) = 4^(3/2)
x = 64^(1/2)
x = 8