Exponent Concepts

From Paper to Pandemic

Have you ever imagined folding a sheet of paper 42 times? If it were possible to do so, its thickness would exceed the distance from Earth to the Moon! This is because each fold doubles the thickness of the paper—this is what we call exponential growth.

Exponential growth occurs when something increases by a constant multiplier in each time interval. In early 2020, the world experienced a real example of exponential growth through the spread of the COVID-19 virus. One infected person could transmit to two people, then four, eight, and so on.

Definition of Exponents

An exponent is a shorthand way to write repeated multiplication. Imagine you are calculating how many people are infected with a virus like COVID-19. At each transmission phase, the number of infected people will increase in an interesting pattern:

This pattern continues, so at phase $n$, the number of infected people can be expressed as $m(n) = 2^n$.

For example, if you want to know how many people are infected at phase 5, you just calculate:

$m(5) = 2^5 = 32$ people.

Meaning of Exponent Notation

An expression with an exponent like $a^n$ has two important components:

Where:

• $a$ is the base - the number that will be multiplied repeatedly
• $n$ is the exponent - indicates how many times the base is multiplied by itself

In general, if $a$ is a real number and $n$ is a positive integer, then:

Important Definitions in Exponents

Here are some important definitions you need to know:

Zero Exponent

For any real number $a$ where $a \neq 0$:

This might seem strange at first, but this definition maintains consistency in the properties of exponents.

Negative Exponents

For any real number $a$ where $a \neq 0$ and a positive integer $n$:

This means a negative exponent equals one divided by the base raised to the same (positive) exponent. This formula is derived from the consistency of exponent properties. To use this formula, you simply flip the base and change the sign of the exponent. Example: $3^{-2} = \frac{1}{3^2} = \frac{1}{9} = 0.111...$

Fractional Exponents

If $a$ is a real number where $a \neq 0$ and $n$ is a positive integer, then:

where $p$ is a positive real number such that $p^n = a$.

The number $a^{\frac{1}{n}}$ is also often called the nth root of $a$. This formula emerges as the inverse of exponentiation. To use it, you need to find the number that, when raised to the power of n, will produce a. Example: $16^{\frac{1}{4}} = 2$ because $2^4 = 16$.

Mixed Fractional Exponents

If $a$ is a real number where $a \neq 0$ and $m,n$ are positive integers, then:

This formula is obtained by combining the concepts of roots and exponents. To calculate it, you must first find the nth root of a, then raise it to the power of m. Example: $8^{\frac{2}{3}} = (8^{\frac{1}{3}})^2 = 2^2 = 4$.

Exponential Functions

Exponential functions have the form $f(x) = a^x$ where $a > 0$ and $a \neq 1$. There are two interesting cases:

1. If $a > 1$, the function will increase (growth)
2. If $0 < a < 1$, the function will decrease (decay)

Exponential functions are very useful in real life because many natural phenomena follow patterns of exponential growth or decay.

Real-Life Applications

Bacterial Growth

One bacterium can divide into two, then four, eight, and so on. If $B_0$ is the initial number of bacteria and each bacterium divides every hour, then the number of bacteria after $t$ hours is:

This formula is obtained because the bacterial population doubles at each time interval. The number 2 represents the growth factor. To use it, multiply the initial amount by 2 raised to the power of the number of intervals that have passed. Example: if there are initially 100 bacteria and they divide every 30 minutes, after 2 hours (4 intervals) there will be $100 \times 2^4 = 100 \times 16 = 1,600$ bacteria.

Virus Spread

The pattern of virus spread, like COVID-19, also often follows an exponential model, especially in the early phase. If one person can transmit the virus to an average of $R$ new people (reproduction number), the number of cases after $n$ transmission cycles can be estimated by:

where $C_0$ is the initial number of cases.

This formula is similar to bacterial growth, but with a multiplier factor $R$ that can vary. This formula is obtained by multiplying the number of cases by $R$ in each transmission cycle. To use it, multiply the initial number of cases by $R$ raised to the power of the number of cycles that have passed. Example: if $R = 2.5$ and there are 10 initial cases, after 3 transmission cycles there will be $10 \times 2.5^3 = 10 \times 15.625 = 156.25 \approx 156$ cases.

Population Growth

To predict future population numbers, an exponential model can be used with the formula:

where $P_0$ is the initial population, $r$ is the growth rate, and $t$ is time (usually in years).

This formula is obtained by adding the growth percentage $r$ to the population at each time interval. The factor $(1+r)$ indicates relative growth. To use it, multiply the initial population by $(1+r)$ raised to the power of the number of time intervals. Example: if the initial population is 1 million people with 2% growth per year, after 10 years the population becomes $1,000,000 \times (1 + 0.02)^{10} = 1,000,000 \times 1.22 = 1,220,000$ people.