Exponents and Bases

Exponential curve

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Identifying the exponent and its base is the prerequisite for simplifying expressions with exponents, but first, it's important to define the terms: an exponent is the number of times that a number is multiplied by itself and the base is the number that is being multiplied by itself in the amount expressed by the exponent.

To simplify this explanation, the basic format of an exponent and base can be written bwherein n is the exponent or number of times that base is multiplied by itself and b is the base is the number being multiplied by itself. The exponent, in mathematics, is always written in superscript to denote that it is the number of times the number it's attached to is multiplied by itself.

This is especially useful in business for calculating the amount that is produced or used over time by a company wherein the amount produced or consumed is always (or nearly always) the same from hour to hour, day to day, or year to year. In cases like these, businesses can apply the exponential growth or exponential decay formulas in order to better assess future outcomes.

Everyday Usage and Application of Exponents

Although you don't often run across the need to multiply a number by itself a certain amount of times, there are many everyday exponents, especially in units of measurement like square and cubic feet and inches, which technically mean "one foot multiplied by one foot."

Exponents are also extremely useful in denoting extremely large or small quantities and measurements like nanometers, which is 10-9 meters, which can also be written as a decimal point followed by eight zeros, then a one (.000000001). Mostly, though, average people don't use exponents except when it comes to careers in finance, computer engineering and programming, science, and accounting. 

Exponential growth in itself is a critically important aspect of not only the stock market world but also of biological functions, resource acquisition, electronic computations, and demographics research while exponential decay is commonly used in sound and lighting design, radioactive waste and other dangerous chemicals, and ecological research involving decreasing populations.

Exponents in Finances, Marketing, and Sales

Exponents are especially important in calculating compound interest because the amount of money that is earned and compounded depends on the exponent of time. In other words, interest accrues in such a way that each time it is compounded, the total interest increases exponentially.

Retirement funds, long-term investments, property ownership, and even credit card debt all rely on this compound interest equation to define how much money is made (or lost/owed) over a certain amount of time.

Similarly, trends in sales and marketing tend to follow exponential patterns. Take for instance the smartphone boom that started somewhere around 2008: At first, very few people had smartphones, but over the course of the next five years, the number of people who purchased them annually increased exponentially.

Using Exponents in Calculating Population Growth

Population increase also works in this way because populations are expected to be able to produce a consistent number more offspring each generation, meaning we can develop an equation for predicting their growth over a certain amount of generations:

c = (2n)2

In this equation, c represents the total number of children had after a certain number of generations, represented by n, which assumes that each parent couple can produce four offspring. The first generation, therefore, would have four children because two multiplied by one equals two, which would then be multiplied by the power of the exponent (2), equalling four. By the fourth generation, the population would be increased by 216 children.

In order to calculate this growth as a total, one would then have to plug the number of children (c) into an equation that also adds in the parents each generation: p = (2n-1)2 + c + 2. In this equation, the total population (p) is determined by the generation (n) and the total number of children added that generation (c). 

The first part of this new equation simply adds the number of offspring produced by each generation before it (by first reducing the generation number by one), meaning it adds the parents' total to the total number of offspring produced (c) before adding in the first two parents that started the population.

Try Identifying Exponents Yourself!

Use the equations presented in Section 1 below to test your ability to identify the base and exponent of each problem, then check your answers in Section 2, and review how these equations function in the final Section 3.

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Exponent and Base Practice

Identify each exponent and base:

1. 34

2. x4

3. 7y3

4. (x + 5)5

5. 6x/11

6. (5e)y+3

7. (x/y)16

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Exponent and Base Answers

1. 34
exponent: 4
base: 3

2. x4
exponent: 4
base: x

3. 7y3
exponent: 3
base: y

4. (x + 5)5
exponent: 5
base: (x + 5)

5. 6x/11
exponent: x
base: 6

6. (5e)y+3
exponent: y + 3
base: 5e

7. (x/y)16
exponent: 16
base: (x/y)

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Explaining the Answers and Solving the Equations

It's important to remember the order of operations, even in simply identifying bases and exponents, which states that equations are solved in the following order: parenthesis, exponents and roots, multiplication and division, then addition and subtraction.

Because of this, bases and exponents in the above equations would simplify to the answers presented in Section 2. Take note of question 3: 7y3 is like saying 7 times y3. After y is cubed, then you multiply by 7. The variable y, not 7, is being raised to the third power.

In question 6, on the other hand, the entire phrase in the parenthesis is written as the base and everything in the superscript position is written as the exponent (superscript text can be regarded as being in parenthesis in mathematical equations such as these).

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Your Citation
Ledwith, Jennifer. "Exponents and Bases." ThoughtCo, Feb. 16, 2021, thoughtco.com/exponents-and-bases-2312002. Ledwith, Jennifer. (2021, February 16). Exponents and Bases. Retrieved from https://www.thoughtco.com/exponents-and-bases-2312002 Ledwith, Jennifer. "Exponents and Bases." ThoughtCo. https://www.thoughtco.com/exponents-and-bases-2312002 (accessed March 27, 2023).