What are Exponents?

The exponent or power or index of a variable/number is the number of times that variable/number is multiplied by itself.

${\text{For example, x}}^3\text{ means that x is multiplied by itself 3 times}\text{.}$

${\text{Thus, x}}^3\text{ = x}\times \text{x}\times \text{x}\text{.}$

Here x is called the base and 3 is called the exponent/power/index.

${\text{Similarly, 2}}^4\text{ = 2}\times \text{2}\times \text{2}\times \text{2 = 16}\text{.}$

Analogy to Exponents

Have you ever played the game Chinese whispers? In that game, a single message is passed around a group of people, one at a time. One person starts and then passes the message to only one person who is next in line, who then passes it to just one other person who is further in line and this continues.

Two people, A and B, start the chain and each person passes on the message to only one more person and so on.

Now think of a variation of this game. Suppose that two people start the game, A and B, just like before. But this time, each person passes on the message to two people, and those two people, in turn, each pass on the message to two more people and so on.

The message in the second case reaches more people in the same number of rounds. This is also often how gossip and rumors spread. Similarly, this is the way exponents work-the constant or variable is multiplied with itself over and over again.

Mathematically speaking, exponents have the power to raise a quantity to a larger degree than simple addition. For instance,

3 + 3 = 6

3 * 3 = 9

Laws of Exponents

The easiest way to calculate multiple exponents would be to write down the variable the same number of times as the power.

${\text{For example, x}}^3{\text{x}}^4={\text{ (xxxx)(xxx) = x}}^7.$

$\begin{array}{l}\text{Naturally, this is not a practical option when the powers are higher}\text{. For}\\ {\text{instance, if the exponential function is shown as x}}^{231}{\text{ or x}}^{54},\text{ it would make}\\ {\text{no practical sense to write x}}^{231}{\text{ times or x}}^{54}\text{ times respectively}\text{.}\end{array}$

This is why we have certain laws of exponents, which simplify calculations and make computation quicker. These laws are as follows:

$1.{\text{ x}}^1\text{ = x}$

(Tip: Use definition and write x one time)

Examples: 23¹ = 23

                (1/5)¹ = 1/5

$2.{\text{ x}}^0\text{ = 1}$

Examples: 4⁰ = 1

               (2/5)⁰ = 1

$3.{\text{ x}}^{-1}\text{ = 1/x}$

Examples: 56^-1 = 1/56

                (1/b)^-1 = b/1 = b

The inverse of a number is a value that when multiplied with the original value gives you the identity value 1. Consider:

${\text{xx}}^{-1}\text{ = x/x = 1}$

$2\times 2^{-1}\text{ = 2/2 = 1}$

The exponent –1 gives us the multiplicative inverse of a variable or constant.

To sum up, positive exponents enlarge the value, negative exponents reduce the value, an exponent of 1 retains the value and an exponent of 0 gives the value 1.

$\text{4}{\text{. x}}^{-\text{n}}{\text{ = 1/x}}^{\text{n}}$

$\begin{array}{l}{\text{Examples: x}}^{-4}{\text{ = 1/x}}^4\\ {\text{ 3}}^{-4}{\text{ = 1/3}}^4\text{ = 1/81}\end{array}$

A positive exponent implies multiplication and thus, a negative exponent implies the opposite, which is division. This law can also be understood as an extension of Law 3.

$\text{5}{\text{. x}}^{\text{m}}{\text{x}}^{\text{n}}{\text{ = x}}^{\text{m+n}}$

${\text{Clearly, x}}^2{\text{x}}^3{\text{ = (xx)(xxx) = x}}^5{\text{ = x}}^{2+3}$

${\text{Similarly, x}}^{34}{\text{x}}^{21}{\text{ = x}}^{34+21}{\text{ = x}}^{55}$

$\text{6}{\text{. x}}^{\text{m}}/{\text{x}}^{\text{n}}{\text{ = x}}^{\text{m-n}}$

${\text{Clearly, x}}^5/{\text{x}}^3{\text{ = xxxxx/xxx = xx = x}}^2{\text{ = x}}^{5-3}$

${\text{Similiarly, x}}^{54}/{\text{x}}^{34}{\text{ = x}}^{54-34}{\text{ = x}}^{20}$

$\text{7}{\text{. (x}}^{\text{m}}{)}^{\text{n}}{\text{ = x}}^{\text{mn}}$

${\text{Clearly, (x}}^2{)}^3{\text{ = (xx)}}^3{\text{ = (xx)(xx)(xx) = x}}^6{\text{ = x}}^{2\times 3}$

${\text{Similiary, (x}}^5{)}^7{\text{ = x}}^{5\times 7}{\text{ = x}}^{35}$

$8.{\text{ (xy)}}^{\text{n}}{\text{ = x}}^{\text{n}}{\text{y}}^{\text{n}}$

${\text{Clearly, (xy)}}^3{\text{ = (xy)(xy)(xy) = xxxyyy = x}}^3{\text{y}}^3$

${\text{Similarly, (xy)}}^7{\text{ = x}}^7{\text{y}}^7$

$\text{9}{\text{. (x/y)}}^{\text{n}}={\text{ x}}^{\text{n}}/{\text{y}}^{\text{n}}$

${\text{Clearly, (x/y)}}^2{\text{ = (x/y)(x/y) = xx/yy = x}}^2/{\text{y}}^2$

${\text{Similarly, (x/y)}}^5{\text{ = x}}^5{\text{/y}}^5$

10. Combining Laws 9 and 4, we also get:

(a/b)^-n = a^-n/b^-n = bⁿ/aⁿ = (b/a)ⁿ

11. Fractional Exponents

An exponent which is a fraction implies a base root of the function.

$x^{\frac{1}{n}}=\sqrt[n]{x}$

${\text{Example: x}}^{1/2}\text{ = }\sqrt{\text{x}}$

Also, note that by combining laws 11 and 7, we get

$x^{\frac{m}{n}}=x^{m\times \frac{1}{n}}={\left(x^m\right)}^{\frac{1}{n}}=\sqrt[n]{x^m}$

Example: $x^{\frac{2}{3}}=x^{2\times \frac{1}{3}}={\left(x^2\right)}^{\frac{1}{3}}=\sqrt[3]{x^2}$

$12.\text{ If x is a positive number greater than 1, then:}$

${\text{x}}^{\text{m}}{\text{ = x}}^{\text{n}}\Rightarrow \text{ m = n}$

${\text{Example: 2}}^{\text{x}}\text{ = 16 }\Rightarrow {\text{ 2}}^{\text{x}}{\text{ = 2}}^4\Rightarrow \text{ x = 4}$

13. If m is non-zero, then

a^m = b^m ⇒ a=b

         Example: a⁵ = 32 ⇒ a⁵ = 2⁵  ⇒ a=2

14. If p, q and r and distinct primes, then:

p^a q^b r^c = p^m qⁿ r^o ⇒ a=m, b=n, c=o

Example: 2^a 3^b 5^c = 1500 = 4 * 3 * 125 = 2² 3¹ 5³   ⇒ a =2, b=1, c=3

Real-Life Applications of Exponents

Exponents have a wide variety of applications in real life, especially in Science. For starters, they make it easier to write many times raised numbers E.g. 0.0000000000001 can be more conveniently written as 10^-13.

Similarly, they help in finding the accurate answer for the value of measurements because the values can be expressed precisely using exponents.

Lastly, specific measurements like the pH value of substances have an exponential equivalent. This is extremely useful for scientists and researchers. Biologists also use exponential models and numbers to track population growth, radioactive decay, and the like. Often, the spread of viruses and communicable diseases is tracked using exponents.

Likewise, economists use exponents to draw out projections for revenue number and income. Marketers may also use strategies to make use of the exponential spread of information to make their product rule or services multiply and known to everyone by word of mouth or sharing on social media.

Common Mistakes

Here are a few calculation errors that one can avoid when solving for exponent:

1. When there is an equation that requires the use of multiple laws of exponent, be careful with the answer and go step by step, preferably inwards to outwards.

e.g. Solve 8^2/3

Step 1: Write the innermost numeral as an exponent

 8^2/3 = (2³)^2/3

Step 2: Use the multiplication law to solve

⇒ 8^2/3 = 2^3*2/3 = 2²

Step 3: Simplify the exponent

⇒ 8^2/3 = 4

2. Be clear with the difference among the following:

$\begin{array}{l}{\text{(a) x}}^2{\text{x}}^3={\text{ x}}^5\\ {\text{(b) (x}}^2{)}^3{\text{ = x}}^6\\ {\text{(c) x raised to the power 2}}^3{\text{ = x}}^8\end{array}$

3. Be clear with the difference between the following:

$\begin{array}{l}{\text{(a) ax}}^{-1}=\text{ a/x}\\ \text{(b) (ax}{)}^{-1}\text{ = 1/ax}\end{array}$

Context and Applications

The topic is useful for K-12, graduate, and postgraduate courses, especially in the following courses:

• B.Sc. Mathematics
• M.Sc. Mathematics

Related Concepts

• Math
• Logarithm
• Algebra
• Fraction
• Advanced Math and Algebra