Understand the Nature of Exponential Functions in 10 Minutes, Auto Expert Mathematician!

Illustration of exponential properties (freepik)

YOGYAKARTA - Do you often find it difficult when dealing with exponentials? Don't worry! Understanding the nature of exponentials is the key to simplifying complicated calculations to be very easy.

Exponents are not just a row of numbers, but a powerful tool in science and technology. Let's dissect each of its properties so that you can apply it correctly and quickly.

What is Exponential?

As reported by VOI from the Wikipedia page, an exponential is a mathematical operation involving two numbers, namely the base number and the exponent. Exponents are usually written in the form of 𝑏𝑛, which is read as "b to the nth power".

Well, in simple terms, exponential means repeated multiplication.

Here's an example:

23=2×2×2=852=5×5=25

That is, an exponential is a real number multiplied by itself as many times as its exponent value.

How to Write and Read Exponents

The exponential itself is written with a small exponent above the base number, for example 34. The way to read it can be various, such as:

"three to the fourth power" "three to the fourth power" "three to the fourth power" (specifically to the second power)

Meanwhile, in programming or computer languages, exponentials are often written as b^n.

Properties of Exponential Functions that Must Be Understood

This is the essence of the discussion, as long as you know, understand the nature of exponentials will make calculations much more efficient, here are some of its properties:

Exponential Multiplication Properties

If the base numbers are the same, then their powers are added: 𝑏𝑛×𝑏𝑚=𝑏𝑛+𝑚

Example:

23×22=23+2=25

Exponential Division Properties

If the base number is the same, then its exponent is reduced, this is the formula 𝑏𝑛:𝑏𝑚=𝑏𝑛−𝑚

Example:

54:52 = 54−2=52

Zero Rank Properties

Every number (other than zero) to the zero power is one, by the formula 𝑏0=1

Example:

70=1

Negative Order Property

A negative exponent means the inverse of a positive exponent, with the formula 𝑏−𝑛=1/𝑏𝑛

Example:

2−3=1/23=⅛

Read also the article which discusses Who Invented the Algorithm? Check out the Story and Background of Al-Khawarizmi Below ​

Fractional Order Properties

Fractional powers are related to roots, use the formula 𝑏1/2=, where this property is very important to understand the relationship between exponentials and roots.

Example:

91/2 == 3

Exponentials with Zero, Negative, and Fractional Powers

Exponential is not only valid for positive integers. Interestingly, with the same rule, exponentials can be extended to:

Zero rankNegative rankFraction rank

All these developments still follow the main rules of exponential properties, especially the properties of summation and subtraction of powers.

A brief history of exponentials

The exponential concept has been known since ancient times. Archimedes used the law of exponents to process large numbers. Muslim scientists such as Al-Khwarizmi also developed the concept of squares and cubes.

Modern exponential notation came into wide use after its introduction by René Descartes, and was refined by Leonhard Euler who extended the use of exponents to non-integer numbers.

The Application of Exponentials in Everyday Life

Exponentials are not just in textbooks. Some examples:

Compound interest calculation in economics Population growth Chemical reactions Waves and physics Digital security systems and cryptography

All of that works with the principle of exponential properties.

Common Mistakes in Using the Exponential Property

Some of the common mistakes are:

Summing the principal number, not the exponent Misunderstanding negative exponents Assuming 𝑏0=0

However, by understanding the basic concept, various errors above can be avoided.

Understanding the nature of exponential properties is actually not difficult if done gradually and logically. With the right concept, exponentials are actually one of the easiest mathematics to master.