BM4.1 – Working with Exponentiation and Power Laws

Chapter 4, Lesson 1

In this lesson students will:

understand how to interpret the relationship between the base and the power of an exponential expression
utilize the power laws and the BAII Plus calculator to simplify or evaluate exponential expressions

Exponentiation & Power Laws

Interpretation of Powers

Let’s begin this lesson by refreshing our memory on how we interpret various expressions involving exponentiation. Every exponential expression involves two parts: the base, and the power. For example in the expression $2^{3}$ , the number $2$ is the base, and the number $3$ is the power. The interaction of the base and the power is a mathematical operation known as exponentiation.

In its simplest sense, we can think of exponentiation as repeated multiplication. This is quite easy to do with whole number bases and powers. For example, we can think of $2^{3} = 2 \times 2 \times 2$ . So we see that the base is the number that is repeatedly multiplied, and the power is the number of times we repeat the multiplication.

When integer exponents are involved, interpretations have to be extended a bit. How do we think about $2^{-3}$ , for instance? Well, if positive powers represent repeated addition, the natural thing to do is interpret a negative power as repeated division. This is where the rule of making the expression a fraction over one with a positive power comes from. Going back to our previous example of $2^{-3}$ , we get

$\begin{aligned} 2^{-3} &= 1 \div 2 \div 2 \div 2 \\ &= \frac{1}{2\times 2\times 2}\\ &= \frac{1}{2^{3}} \end{aligned}$

Finally, provided that the base number is not zero, any base raised to the power of zero is equal to one. For example, $45^{0} = 1$ , $(-210)^{0} = 1$ , and $x^{0} = 1$ (provided $x\neq 0$ ). We will give a general proof of this concept in the next section.

Power Laws

The five major power laws can be summarized in English in the following way:

A multiplication of similar bases allows us to add the powers: $a^{m}a^{n}=a^{m+n}$ .
A division of similar bases allows us to subtract the powers: $\displaystyle \frac{a^{m}}{a^{n}} = a^{m-n}$ .
An exponential expression raised to a power can be simplified by multiplying the powers: $(a^{m})^{n} = a^{m \times n}$ .
Exponentiation can be distributed to all terms connected by multiplication: $(a\times b)^{m} = a^{m} \times b^{m}$ .
Exponentiation can be distributed to the numerator and denominator of a fraction: $\displaystyle \left( \frac{a}{b} \right)^{m} = \frac{a^{m}}{b^{m}}$ .

Utilizing property two, we can give insight into why $a^{0}=1$ , provided $a\neq 0$ . If $m=n$ in property two, we have

$\begin{aligned} \frac{a^{m}}{a^{m}} &= a^{m-m}\\ &= a^{0}. \end{aligned}$

Of course, since $a \neq 0$ , we also see that $\displaystyle \frac{a^{m}}{a^{m}}$ is a fraction in which the numerator and denominator are the same number. This means $\displaystyle \frac{a^{m}}{a^{m}} = 1$ , showing us that $\displaystyle a^{0} = \frac{a^{m}}{a^{m}} = 1$ , provided $a \neq 0$ (there is no division by zero).