Algebra
Worked Examples
Exponents
Part 1: The Exponential Form - $a^n$
For some integerThe integers are the infinite set of numbers $\ldots -2, -1, 0, 1, 2, \ldots$They consist of the natural numbers, $1,2,3,4,\ldots$, plus 0, plus the negative natural numbers. $a$, and natural number $n$, where $n \geq 1$: $$a^n = \underbrace{a\cdot a \cdots \cdots a \cdot a}_{n \textrm{ times}}$$ Here, $a$ is called the base, and represents any number being raised to the power of/exponent $n$. We can see that exponential form of this kind just implies repeated multiplication! Below are some exponent rules that will be quite useful as we move forward.
| Properties of Exponents | |
|---|---|
| 1. | $a^m \cdot a^n = a^{m+n}$ |
| 2. | $\frac{a^m}{a^n} = a^{m-n}$ |
| 3. | $ (a^m)^n = a^{mn} $ |
| 4. | $(ab)^n = a^n b^n$ |
| 5. | $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ |
Now, we will make a few observations that will uncover a couple more properties. We’ve always been taught that for any number $a \neq 0, \; \frac{a}{a}=1$. However, by Property 2, we also have the following: $$1 = \frac{a}{a} = \frac{a^1}{a^1} = a^{1-1} = a^0$$ As a consequence, we then have: $$\frac{1}{a^n} = \frac{a^0}{a^n} = a^{0-n} = a^{-n} $$  This gives the following additional Properties:
| Properties of Exponents (Continued) | |
|---|---|
| 6. | $a^0 = 1$ |
| 7. | $a^{-n} = \frac{1}{a^n}$ |
Important observations:
- The exponent $0$ makes a number equal to $1$, not $0$!
- A negative exponent simply means to take the reciprocal of a number; it never changes its sign!
Part 2: Rational Exponents - exponential form $a^{\frac{m}{n}}$
Definition: $$ \sqrt[n]{a} = a^{\frac{1}{n}}$$ is called “the nth root of $a$” and $\sqrt[n]{a}$ is known as radical form.
Note:
if $n$ is even, then $a \geq 0$.
To understand what this expression means, we just need to satisfy the following: $$\sqrt[n]{a} = k \qquad \textrm{is equivalent to}\qquad k^n = a. \qquad \textrm{ Find }k. $$
Example. $\sqrt[4]{16}$
Solution. $\sqrt[4]{16}=k $, such that $k^4 = 16$. Therefore, $k = \sqrt[4]{16} = 2$Example. $\sqrt[3]{-27}$
Solution. $\sqrt[3]{-27}=k$, such that $k^3 = -27$.Therefore, $k = \sqrt[3]{-27} = -3$.
Example. $\sqrt[4]{-256}$
Solution. $\sqrt[4]{-256} = k$, such that $k^4 = -256$.But, no such $k$ exists since any real number raised to the fourth power must be positive! Therefore, there is no solution!
Note:
Now it’s clear why $a$ must be positive if $n$ is even
Combining Definition 1 and Property 3: $(a^u)^w = a^{uw}$, we have: $$ a^{\frac{m}{n}} = \left\{ \begin{array}{ll} (a^m)^{\frac{1}{n}} = \sqrt[n]{a^m} & \textrm{ Form 1} \\ \left(a^{\frac{1}{n}}\right)^m = (\sqrt[n]{a})^m &\textrm{ Form 2} \end{array}\right. $$ Clearly Form 1 and Form 2 are equivalent. However, Form 2 is particularly useful because it allows you to take the $n$th root of a smaller number!
Example.
$$\begin{align*} 81^{\frac{3}{4}} = \left( 81^{\frac{1}{4}} \right)^3 &= (\sqrt[4]{81})^3 \qquad\qquad \textit{ ...and we know that }\sqrt[4]{81}=3 \\ &= (3)^3 \\ &= 27. \end{align*}$$Now we’ll look at the case when $m=n$.
Example.
It’s clear that $\sqrt[4]{(2)^4} = (\sqrt[4]{2})^4 = (2)^{\frac{4}{4}} = (2)^1 = 2$.
Is this true for any $a$ and $n$?
Consider $\sqrt[4]{(-3)^4}$. Is the solution $−3$? No!
Notice that if $n$ is even, $(−1)^n = 1$. Consequently, $(-a)^n = (-1)^n(a)^n = (a)^n$ for any positive $a$.
Therefore, $\sqrt[4]{(-3)^4} = \sqrt[4]{(3)^4} = 3^{\frac{4}{4}} = 3^1 = 3$.
This uncovers a very important property for any integer $a$: $$\sqrt[n]{a^n} = \begin{cases} a & \text{ if $n$ is odd} \\ |a| & \text{ if $n$ is even} \end{cases} $$ Below are some worked examples that combine the Properties of Exponents, as well as Definition 1 and Property 1:
Remember that there may be several different ways to arrive at these final solutions!
Simplify the following expressions and eliminate all negative exponents (if necessary):