Real Number Exponents

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Pre-Calculus $\longrightarrow$

Exponential Relationships $\longrightarrow$

Understanding Variable Exponents

Objectives

Quickly recap all previously learned exponent rules
Make connections between rational exponents, which we've seen, and terminating decimal exponents, which we haven't yet seen
Understand how irrational exponents work and what they mean, even though they are not as intuitive

Lesson Description

Building up to the next several lessons on exponential functions and relationships, we need to understand that we can let any real number be an exponent, from integer and fraction exponents that we have already seen, to irrational numbers like $\sqrt{2}$ and $\pi$.

Practice Problems

Practice problems and worksheet coming soon!

Exponents on Steroids

As you progress through Algebra, exponents become much more complicated than they seemed when you were first introduced to them. The first milestone complication happens when we need to apply exponent rules to numbers and variables, such as what happens when you multiply $x^{4}$ by $x^{12}$. Later on, we combined some knowledge of roots and radicals with that of exponents, and introduced the idea of fraction exponents, such as $x^{4/3}$.This lesson will open up Pandora's Box all the way and consider exponents of any type of number - including terminating decimals, repeating decimals, and even irrational numbers like $\pi$.But first, let's make sure we're up to speed on everything exponents up to this point.

Exponent Recap

Let's quickly look at what we know about exponents.First, we were introduced to exponents » as repeated multiplication. That is indeed what exponents are when they are integers - for example, $3^5$ equals $3 \cdot 3 \cdot 3 \cdot 3 \cdot 3$. This will always be a valid statement and will always be a useful perspective, since we commonly work with integer exponents.Later on, we learned some exponent shortcut rules » for simplifying and working with expressions that contain multiple appearances of exponents, as well as rules for zero and negative exponents ». Specifically, make sure all of the following are familiar:$$x^{a} \cdot x^{b} = x^{a+b}$$$$\frac{x^a}{x^b} = x^{a-b}$$$$\left(x^a\right)^b = x^{ab}$$$$x^{-a} = \frac{1}{x^a}$$$$x^{0} = 1, \; \mathrm{for} \; x \ne 0$$Finally, in Algebra Two, we learned that exponents could be fractions » and gained an understanding of what fraction exponents actually mean.Recall that $x^{a/b}$ tells us to raise $x$ to the $a$ power and then take the $b$-th root (or vice versa - we also learned that the order doesn't matter).For example, $64^{2/3}$ is equal to $16$, because the cube root of $64^2$ is $16$ (or once again, the square of $\sqrt[3]{64}$ is also $16$).Now that we're caught up on exponents up to this point, let's do more.

Decimal Exponents

Decimal exponents are actually not a far stretch from fraction exponents in most cases. As we've learned over the course of our Algebra journey, decimals come in three major flavors: terminating, repeating, and non-terminating non-repeating.Terminating Decimal ExponentsDue to the structure and definition of decimals, terminating decimals can always be written as fractions with a denominator of a power of $10$ (and possibly reduced to a different denominator).Examples of this include$$0.47 = \frac{47}{100}$$$$0.6 = \frac{6}{10} = \frac{3}{5}$$$$0.837629 = \frac{837,629}{1,000,000}$$Simply stated, if a decimal can be written as an equivalent fraction, then we can use what we already know about fraction exponents to deal with decimal exponents.For example, since $0.6$ equals $3/5$, we can say$$x^{0.6} = x^{3/5} = \sqrt[5]{x^3}$$Repeating DecimalsRepeating decimals require more work but ultimately give us the same end result as terminating decimals. This is because both terminating and repeating decimals represent the same type of real number - a rational number, which by definition can always be represented as a simple fraction.If $0.\bar{3}$ is really $1/3$, then$$x^{0.\bar{3}} = x^{1/3} = \sqrt[3]{x}$$Again, this is no different from what we stated a minute ago about terminating decimals. The biggest difference is that repeating decimals can be a pain to convert to rational form.These days we're often allowed to rely on technology for this conversion - especially in advanced Algebra and Pre-Calculus courses. On the TI-84, you can have it convert a repeating decimal to a fraction in two super quick steps.

First, enter twelve or more characters of repeating pattern into the calculator.
Then, press Math $\longrightarrow$ Enter $\longrightarrow$ Enter. This works because the [ $\triangleright$ Frac ] option is the first choice in the Math menu.

Just make sure you enter enough digits. If you only enter $0.333$, your calculator will convert it to the fraction $333/1000$. If you enter twelve or more $3's it will decide that you're actually trying to decode a repeating decimal.It's beyond uncommon, but if you have to do any work to convert repeating decimals to fractions without the aide of a calculator, make sure to check out the lesson on converting repeating decimals algebraically ».Irrational DecimalsDecimals that never end and don't repeat a pattern represent irrational numbers. Unlike rational numbers, irrational ones cannot be expressed as a simple fraction. Some examples of irrational numbers that we've encountered along the way are $\pi$ or $\sqrt{2}$. Your calculator will give you decimal representations of these numbers to as many decimals as you need when asked, but the true decimal representations of these numbers never end.Without a fraction to convert to, irrational number exponents cannot be dealt with using knowledge we currently have.

Irrational Exponents

We need to allow for the possibility that an exponent could be any real number so that we can begin to allow exponents to be variables. In order to do that, we cannot settle for allowing only rational numbers to be exponents, as the set of real numbers consists of the union of the sets of both rational and irrational numbers.We can intuitively understand why the results we get are what they are, but it's important to know right away that there is no perfect conceptual translation of irrational exponents to roots and powers in the same way that exists for fraction exponents.Let's start with an example.

Example 1Find two expressions that use rational exponents such that $2^{\pi}$ is in between the two expressions.$\blacktriangleright$ This question is intentionally somewhat open-ended to illustrate a few new ideas.First, if you grab your calculator and ask it for the decimal approximation of $2^{\pi}$, it gives us$$2^{\pi} \approx 8.824977827$$Nine decimal places is a strong approximation for most practical purposes, so we'll loosely refer to this result as the "true" value.Recall that $\pi$ is approximately $3.1415927$. Recall also that as we raise $2$ to bigger and bigger powers, the result grows. Specifically we know that $2^2 = 4$, $2^3 = 8$, $2^4 = 16$, etc. Putting this all together, it is reasonable to make two proposals:1) $2^{3.1} \lt 2^{3.2}$and furthermore,2) $2^{3.1} \lt 2^{\pi} \lt 2^{3.2}$What we should understand here is that while $2^{3.1}$ can be written as $\sqrt[10]{2^{31}}$ and $2^{\pi}$ cannot, both results follow a logical progression in that $2^{\pi}$ is in between $2^{3.1}$ and $2^{3.2}$ due to the fact that $\pi$ is in between $3.1$ and $3.2$, even though $2^{\pi}$ doesn't have a clean interpretation in terms of roots and powers.Therefore, to answer the question, we can say that $2^{3.1}$ and $2^{3.2}$ are two expressions that use rational exponents and also have $2^{\pi}$ in between them.

You Should Know

We could have found "tighter" boundaries to contain $2^{\pi}$ between simply by using more decimal places. For example, the same steps and logic above could lead us to state that $2^{\pi}$ is between $2^{3.1415}$ and $2^{3.1416}$, which is a narrower corridor.

Working with Functions

As an introduction to the idea of having variables in the exponent rather than the base, consider the following function:$$f(x) = 4^{x}$$One reason we want to understand why both rational and irrational exponents are allowable is so that we can examine functions like this and correctly identify their domain ». As we have been discussing, both rational and irrational numbers are allowed to be exponents, so the domain of this particular function is all real numbers ($x \in \mathbb{R}$).Additionally, we should be prepared to take a function like this and evaluate it at various specific values of $x$.

Example 2For the function $f(x) = 3^x$, evaluate the following:$f(1)$$f(0)$$f(\sqrt{3})$$f(-2)$$\blacktriangleright$ Part of what we learn here is the intuition about what we could and could not do without a calculator. We can actually be expected to calculate each of these quantities calculator-free, except for the third one.$$f(1) = 3^{1} = 3$$$$f(0) = 3^{0} = 1$$$$f(-2) = 3^{-2} = \frac{1}{9}$$Now, the third quantity we were asked for literally can't be simplified once plugged in.$$f(\sqrt{3})=3^{\sqrt{3}}$$However, we can get a decimal approximation with a calculator, if needed.$$3^{\sqrt{3}} \approx 6.705$$

The key takeaway is that exponents can be any real number, not just integers and fractions. We can't interpret irrational exponents the same way that we can for rational ones, but that's not unusual since irrational numbers have a never-ending and never-repeating decimal expansion. But we can get decimal approximations to things like $5^{\sqrt{3}}$ just like we can get decimal approximations to things like $\sqrt{3}$. And when we prefer exact answers we just leave it be.

Put It To The Test

As you become an expert on exponential relationships over the next few lessons, these basic practice problems will quickly become "too easy" to be included on unit tests. However, at this point as you are starting out with this unit, it's important that everything we are saying above is well-understood. Make sure you don't have any hang-ups with the following extra practice problems.

Example 3Find two rational exponent expressions that bound the value of $4^{\sqrt{10}}$.

Show solution

$\blacktriangleright$ Since $\sqrt{10}$ is between $3$ and $4$ (since $\sqrt{9}$ is $3$ and $\sqrt{16}$ is $4$) then we know that $4^{\sqrt{10}}$ must be between $4^3$ and $4^4$. These are the two expressions we can turn in as our answer.If we had a calculator, which can tell us that $\sqrt{10} \approx 3.162$, we could be even more precise and pick something like $4^{3.16}$ and $4^{3.17}$ as our answers. Harder questions will require this kind of precision by asking for two numbers such that the error from the true value is very small, say less than $1/100$-th.

Example 4With the given function, evaluate the exact result at each specified $x$ values, and then use a calculator to evaluate a decimal approximation to those function values.$$f(x) = 9^{x}$$for $x \in {0.5,\; 1,\; 2,\; \pi,\; 4}$.

Show solution

$\blacktriangleright$ Except for the $x$ value of $\pi$, these are all non-calculator computations.$$f(0.5) = 9^{1/2} = 3$$$$f(1) = 9^1 = 9$$$$f(2) = 9^2 = 81$$$$f(4) = 9^4 = 6561$$Now, when $x=\pi$, we should turn in the exact answer as instructed, in addition to the decimal approximation (the previous four answers do not need any approximating since they are exact integers):$$f(\pi) = 9^{\pi} \approx 995.04$$

Lesson Takeaways

Review and re-master all exponent rules from Algebra
Understand the connection between decimal exponents and fraction exponents
See how irrational exponents fit into the grand scheme of numbers similar to how irrational numbers fit in between rational ones
Begin to work with functions that have variable exponents and domains of all real numbers