# 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.