# Algebra

## 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):

### Example. $\left( \frac{100}{9} \right)^{-\frac{3}{2}}$

Solution: \begin{align*} \left(\frac{100}{9} \right)^{-\frac{3}{2}} &= \left(\frac{9}{100} \right)^{\frac{3}{2}} \\ &= \frac{9^{\frac{3}{2}}}{100^{\frac{3}{2}}} \\ &= \frac{(\sqrt{9})^3}{(\sqrt{100})^3} \\ &= \frac{3^3}{10^3} \\ &= \frac{27}{1000} \end{align*}

### Example. $\frac{(8st^{-4})^{1/3}}{(27s^{1/2}t^{-4})^{2/3}}\left(\frac{4t^{-1/3}}{3s^{-2}}\right)^{-2}$

Solution. \begin{align*} \frac{(8st^{-4})^{1/3}}{(27s^{1/2}t^{-4})^{2/3}}\left(\frac{4t^{-1/3}}{3s^{-2}}\right)^{-2} &= \frac{8^{1/3}\cdot s^{1/3}\cdot t^{-4/3}}{27^{2/3}s^{2/6} t^{-8/3}}\cdot \frac{4^{-2}t^{2/3}}{3^{-2}s^4} \\ &= \frac{\sqrt[3]{8}\cdot s^{0}\cdot t^{4/3}}{(\sqrt[3]{27})^2}\cdot \frac{3^{2}t^{2/3}}{4^{2}s^4} \\ &= \frac{2\cdot 1 \cdot t^{4/3}}{(3)^2}\cdot \frac{9t^{2/3}}{16 s^4} \\ &= \frac{2 \cdot 9 \cdot t^{6/3}}{9\cdot 16\cdot s^4} \\ &= \frac{t^2}{8s^4} \end{align*}

### Example. $\sqrt[3]{a^2b}\sqrt[3]{64a^4b}$

Solution. \begin{align*} \sqrt[3]{a^2b}\sqrt[3]{64a^4b} &= (a^2b)^{1/3}(64a^4b)^{1/3} \\ &= (a^2b \cdot 64a^4b)^{1/3} \\ &= (64a^6b^2)^{1/3} \\ &= (64)^{1/3} \cdot a^{6/3} \cdot b^{2/3} \\ &= \sqrt[3]{64} \cdot \left( \sqrt[3]{a^3} \right)^2 \left( \sqrt[3]{b^2} \right) \\ &= 4 \cdot a^2 \left( \sqrt[3]{b^2}\right) \end{align*}

### Example. $18\sqrt{5}+2\sqrt{12}-2\sqrt{245}-\sqrt{75}$

Solution. \begin{align*} 18\sqrt{5}+2\sqrt{12}-2\sqrt{245}-\sqrt{75} &= 18\sqrt{5}+2\sqrt{4\cdot 3}-2\sqrt{49 \cdot 5}-\sqrt{25\cdot 3} \\ &= 18\sqrt{5}+2\sqrt{4}\sqrt{3}-2\sqrt{49}\sqrt{5}-\sqrt{25}\sqrt{3} \\ &= 18\sqrt{5}+2(2)\sqrt{3}-2(7)\sqrt{5}-(5)\sqrt{3} \\ &= 18\sqrt{5}+4\sqrt{3}-14\sqrt{5}-5\sqrt{3} \\ &= 4\sqrt{5}-\sqrt{3} \end{align*}

### Example. $(x^2+2x+1)^{1/4}$

Solution. \begin{align*} (x^2+2x+1)^{1/4} &= \left((x+1)^2\right)^{1/4} \\ &= \left(\left((x+1)^2\right)^{1/2}\right)^{1/2} \\ &= \left(\sqrt{(x+1)^2}\right)^{1/2} \\ &= (|x+1|)^{1/2} \\ &= \sqrt{|x+1|} \end{align*}

### Example. $\sqrt[4]{x^4y^2z^2}$

Solution. \begin{align*} \sqrt[4]{x^4y^2z^2} &= \sqrt[4]{x^4} \sqrt[4]{(yz)^2} \\ &= |x|\sqrt{\sqrt{(yz)^2}} \\ &= |x|\sqrt{|yz|} \\ \end{align*}