c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15^t can be rewritten as (1.15^1⁄₁₂)^12t ≈ 1.012^12t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

At this point, the students should have a firm understanding of the concept of an exponent—a power to which some other quantity is raised. Up until this point, however, exponents probably have always been numbers. Students may not be aware that exponents can be variables, too.

Consider an expression like rⁿ. The r is identified as the base of the expression, while the n is identified as the exponent. Since both r and n are variables, this is a more general type of expression involving exponents than something like r² or x¹⁰, where the base is the variable and the exponent is a number. These general expressions are known as (surprise, surprise) exponential expressions, while the others are usually referred to as power law expressions.

Students should know that the rules for manipulating power law expressions apply for exponential expressions too. The following table summarizes those rules:

In multiplying and dividing exponential expressions, just as in power law expressions, the bases have to be the same. We can use the rules for exponential expressions to transform them into alternate forms, allowing us to interpret the expressions in different ways.

Everyone loves earning money, so let's find out ways to do that. No, we don't mean playing the lotto, because that's all chance and probability. We want exponential growth, so we can earn some dinero the good old fashioned way: investing.

The compound interest expression allows us to calculate the new value of an investment after a certain period of time has elapsed if interest is earned at a certain rate for each period of time. It sounds really complicated, but it looks pretty simple: P(1 + r)ⁿ. In this expression, P is the initial amount or present value, r is the interest rate for each period expressed in decimal form, and n is the number of periods over which the new value is to be calculated.

Let's focus on the second factor of that expression: (1 + r)ⁿ. As an example, let's say we open a savings account which offers a rate of 15% per year. In other words, our money grows by 15% each year. This is represented by the mathematical expression (1 + 0.15)ⁿ = (1.15)ⁿ, where n is the number of years over which we allow this account to grow.

What if we were thinking about switching banks, and they offered a savings account, but expressed their rates on a per month basis rather than a per year basis? Obviously, we want to get the highest rate we can, since that's how we make more money. How could we compare the two?

We use the laws of exponentials. If we want our periods to be in months, we have to transform our exponential expression to be on a per month basis. There are 12 months in a year. So, if the account accrues interest for n years, it accrues interest for 12n months. The exponent of our transformed expression must be 12n.

We can't just change the exponent without also changing the part inside the parentheses, though, because we would be creating a completely different expression. To balance multiplying the exponent by 12, we have to divide any exponent inside the parentheses by 12. Even though it isn't written, there is an exponent there, since 1.15¹ = 1.15. So, the new exponent inside the parentheses must become , and the transformed exponential expression is .

Using the rule that when raising an exponential expression to a power you multiply the exponents, the students can verify that we get the original expression back. We use a calculator to evaluate . The final expression, in terms of monthly periods is (1.012)¹²ⁿ = (1 + 0.012)¹²ⁿ. If the annual rate is 15%, the equivalent monthly rate is approximately 1.2%, and we can now compare this to what the other bank offers.