Rational Exponents

In a previous lesson, you learned how to define the power $a^d$ when $d$ was a negative integer, by using the laws of exponents. In this lesson, you will see that a similar process can be used to define $a^d$ when $d$ is rational: that is, to define $a^{m∕n}$ when $m$ and $n$ are integers with $n ≠ 0$.

The graph of $y=a^x$, where $x$ is an integer, is shown on the grid to the left. If we want to define $a^x$ when $x$ is not an integer, it will mean filling in the gaps in this graph.

When $a$ is positive, do the points look like they could easily be connected to form a single smooth curve?
When $a$ is negative, do the points look like they could easily be connected to form a single smooth curve?

This observation suggests that, if $d$ is not an integer, it will only make sense to define $a^d$ when $a$ is positive. The next few questions will show that this is true.

We’ll start by looking at what happens when you square $a^{1∕2}$. If it is to obey the laws of exponents, then we must be able to say:

$$ (a^{1∕2})^2 = a^{1∕2} ⋅ a^{1∕2} = a^{1∕2+1∕2} = a^1 = a $$

That is, if you square $a^{1∕2}$, you should get $a$. In other words, $a^{1∕2}$ should be a square root of $a$.

If $a$ is negative, does it have a real square root?
If $a$ is negative, does it make sense to define $a^{1∕2}$?

Because we want $a^{1∕2}$ to fill in a gap in the graph to the left, we’ll define it to be the non-negative square root of $a$. That is:

Whenever $a ≥ 0$, $a^{1∕2}=√a$.

What is $25^{1∕2}$?

The quantity $a^{1∕n}$ can be defined in the same way as $a^{1∕2}$. By using the law that $(a^c)^d=a^{cd}$, we can see that:

$$ (a^{1∕n})^n = a^{(1∕n) ⋅ n} = a^1 = a $$

and so $a^{1∕n}$ must be a number whose $n$th power is $a$.

Notice that $4^5=1024$. What is $1024^{1∕5}$?
What is $8^{1∕3}$?

If $a$ is non-negative and $n$ is a positive integer, then in fact there is exactly one non-negative number whose $n$th power is $a$.

Suppose $a$ is a real number and $a ≥ 0$, and $n$ is an integer and $n > 0$. Then the non-negative number $b$ with $b^n=a$ is called the $n$th root of $a$, written $√^n{a}$. In this case, we also define $a^{1∕n}=√^n{a}$.

If you know what $a^{1∕n}$ means, you can use that to find $a^{m∕n}$, in one of the following two ways:

$$ a^{m∕n} = a^{m ⋅ 1∕n} = (a^m)^{1 ∕ n} = √^n{(a^m)} $$ $$ a^{m∕n} = a^{(1∕n) ⋅ m} = (a^{1 ∕ n})^m = (√^n{a})^m $$

Use the two formulas above to compute $32^{2∕5}$ in two different ways. (Remember from a previous problem that $1024^{1∕5}=4$.)

Do these two methods of computing $32^{2∕5}$ produce the same result?

Using either of these methods, and scratch paper if necessary, compute the rational powers in the table below.

Some values of $y=2^x$ are plotted on the grid to the left. Using the formula $a^{m∕n}=√^n{(a^m)}$, rewrite each missing power of 2 in the table below as an $n$th root, and then compute that root using a calculator. Round your results to four decimal places. Your answers will also be plotted on the grid to the left.

$x$$2^x$$y$
Do the fractional powers of 2 you computed look like they lie on the same smooth curve as the integer powers?

Click to graph the equation $y=2^x$.

The equations $y_1=2^x$ and $y=a^x$ are graphed on the grid to the left, with a slider for $a$.

Set $a$ to a value greater than 1. As $x$ increases, does $y$ increase or decrease?
Now set $a$ to a value less than 1. As $x$ increases, does $y$ increase or decrease?
Set $a$ to 1.5. Where $x$ is positive, does the graph of $y=1.5^x$ lie above or below the graph of $y_1=2^x$?
Now set $a$ to 3. Where $x$ is positive, does the graph of $y=3^x$ lie above or below the graph of $y_1=2^x$?
Now set $a$ to $0.5$, so the blue curve is the graph of $y=0.5^x$, or $$y=(1/2)^x$$. You should be able to reflect the grid across a line to swap the red and blue curves. Is that line the $x$-axis or the $y$-axis? The -axis.

So far, we’ve only defined $a^d$ when $d$ is rational. What if $d$ is irrational? Then we can define $a^d$ by using rational approximations. For example, $√2=1.4142...$ is approximated more and more closely by the rational numbers 1.4, 1.41, 1.414, 1.4142, and so on. So you can approximate $3^{√2}$ more and more closely by the numbers

$$ 3^1.4,\,3^{1.41},\,3^{1.414},\,3^{1.4142},... $$

This defines $a^d$ for any positive real number $a$ and any real number $d$. When it is defined in this way, the graph of $y=a^x$ forms a single smooth curve, as you can see on the grid to the left.

We defined $a^{m∕n}$ so that it would satisfy the rule $(a^c)^d = a^{cd}$ when $c$ or $d$ was equal to $$1/n$$. In fact, all of the fundamental laws of exponents hold for any real exponents, whenever the bases of those powers are positive.

Whenever $a$ and $b$ are positive, $$ a^c a^d = a^{c + d} $$ $$ (a b)^d = a^d b^d $$ $$ (a^c)^d = a^{cd} $$ for any real numbers $c$ and $d$.

Assuming that $a$ and $b$ are positive numbers, simplify the expressions in the table below. Write all fractions in lowest terms.