Exponent – Definition and Rules

Numbers are often read in words. For example, numbers can be written in terms of hundreds, thousands, lakhs, crores, etc. But how can we read numbers with a large number of digits? Let us take an example.

The weight of the sun is 1.989 × 10³⁰ kg. You can see that here we have taken the help of exponents to express such a large number. Thus, to deal with these types of numbers, we use exponents. In this article, we will walk you through the introduction of exponents, rules, and examples.

Definition of an Exponent

The definition of an exponent in math can be given as the number of times a number is used in multiplication. It can also be defined as the method of expressing large numbers in terms of powers. An exponent refers to how many times a number is multiplied by itself.

It is written as a small number to the right and above the base number.

In this example: 6² = 6 × 6 = 36. (The exponent “2” says to use the 6 twice in multiplication.)

The symbol that we use for representing the exponent is ^. This symbol (^) is called a carrot. Let us look at another example, 8 raised to 2 can be written as 8² or 42. Thus, 8² = 8 × 8 = 64. It has other names like index or power.

Exponent Rules

There are different laws of its that are described based on the powers they bear.

• Multiplication Law: Bases – multiplying the like ones; add the exponents and keep the base the same. When bases are raised with power to another, multiply the exponents and keep the base the same.
• Division Law: Bases – dividing the like ones; subtract the exponent of the denominator from the exponent of the numerator Exponent and keep the base the same.

Let ‘a’ be any integer, or it can also be a decimal number, and ‘m’, and ‘n’ are positive integers that represent the powers to the bases such that the above laws can be written as:

a^m. aⁿ = a^m+n

(am)ⁿ = aⁿmⁿ

(ab)ⁿ = aⁿbⁿ

(a/b)ⁿ = aⁿ/bⁿ

a^m/aⁿ = a^m-n

a^m/aⁿ = (1/a)^n-m

These laws can also be referred to as the properties of its. When you deal with complex algebraic expressions and need to write large numbers understandably, these laws help you. Now let us look at these laws in detail.

Product Rules

These rules are universally applicable to all mathematical calculations. These product rules will help to speed up your calculations in various competitive exams. You must make sure to practice them thoroughly and solve as many problems as you can till your exam day. 

• Product rule with same base

a n ⋅ a m = a n+m2^3 2^4 = 2^(3+4)  = 2^7 = 2⋅2⋅2⋅2⋅2⋅2⋅2 = 128

• Product rule with same exponent

aⁿ ⋅ bⁿ = (a ⋅ b)ⁿ

3² ⋅ 4² = (3*4)² = 12² = 12*12 = 144

• Exponents quotient rules

Quotient rule with same base

aⁿ / a^m = a^n–m

Example:

2⁶ / 2³ = 2^6-3 = 2³ = 2*2*2 = 8

Quotient rule with same exponent

aⁿ / bⁿ = (a / b)ⁿ

Example:

4³ / 2³ = (4/2)³ = 2³ = 2*2*2 = 8

Exponents Power Rules

You should practice the following exponent power rules to master the concept and score more in the exams. Follow the techniques and you will understand the concept within no time. 

• Power rule I

(aⁿ) ^m = a ^n⋅m

Example:

(2*2)² = 2^(2*2) = 2⁴ = 2*2*2*2 = 16

• Power rule II

_a n^m _{= a} (n^m)

• Power rule with radicals

^m√(a ⁿ) = a ^n/m

^Example:

^{2√(26) = 26/2 = 23 = 2⋅2⋅2 = 8}

Negative Exponents Rule

b^-n = 1 / bⁿ

^Example:

^{2^-3 = ½^3 = 1/(2⋅2⋅2) = 1/8 = 0.125}

Zero Rules

b⁰ = 1

0ⁿ = 0 , for n>0

One Rules

b¹ = b

1ⁿ = 1

Multiplying Exponents

Do you have trouble multiplying large values of its? Then, after reading this section, there won’t be any problem. Multiplying exponents can be defined as any two expressions where exponents are multiplied. There are various rules associated when it comes to the multiplication of exponents. These rules depend upon the base and the power. You can sometimes find it difficult to understand because of different bases, negative exponents, and non-integer exponents. 

Before learning more about multiplying exponents, let us revise the meaning of exponents. An exponent can be defined as the number of times a quantity is multiplied by itself. For example, when 3 is multiplied twice by itself, it is expressed as 3 × 3 = 3^2. Here, 3 is the base, and 2 is the power or exponent. It is read as “3 raised to the power of 2”.

Now, let us dive into the meaning and rules associated with multiplying exponents. We will walk you through the different cases with the help of examples to understand the concept in a better way.

Multiplying Exponents with the Same Base

Now, let us consider two expressions with the same base, that is, a^n and a^m. Here, the base is ‘a’. When the terms with the same base are multiplied, the powers are added, i.e.,

 a^m × aⁿ = a^(m+n)

Rules for Multiplying Exponents with a Different Base

We can multiply the expressions using rules when two numbers or variables have different bases. When multiplying exponents with a different base, we have two scenarios:

• When the bases are different, and the powers are the same.

The first scenario is when two expressions have different bases and the same power. Let us look at aⁿ and bⁿ. Here, the bases are a and b, and the power is n. When multiplying exponents with different bases and the same powers, we first multiply the bases. It can be written mathematically as aⁿ × bⁿ = (a × b)ⁿ.

• When the bases and powers are different.

The second scenario is when two expressions have different bases and powers, like aⁿ and b^m. Here, the bases are a and b. The powers are n and m. When the expressions with different bases and powers are multiplied, each expression is evaluated separately and then multiplied. We can mathematically write it as aⁿ × b^m = (aⁿ) × (b^m)

Example: Multiply the expressions: 3³ × 2⁴.

Solution: Here, the bases and the powers are different. Therefore, each term will be solved separately. 3³ × 2⁴ = 27 × 16 = 432.

Negative Exponents

Next, we are looking at negative exponents. They tell us that the power of a number is negative, and it applies to the reciprocal of the number. We know that an exponent refers to the number of times a number is multiplied by itself. For example, 2³ = 2 x 2 x 2 . In the case of positive exponents, we easily multiply the base by itself, but things are done differently for negative numbers as exponents.

A negative exponent can be well defined as the multiplicative inverse of the base, raised to the power opposite to the given power. In other words, we can first write the reciprocal of the number and then solve it like positive exponents. For example, (2/3)^-2 can also be written as (3/2)².

Observe the table. Here are some examples that will help you understand the concept better.

Negative Exponent	Result
2-1	1/2
2-3	1/23 = 1/8
x-6	1/x6

Fractional Exponents

Lastly, we will learn about fraction exponents. A fractional exponent is when an exponent of a number is a fraction. It shows the number of times a number is replicated in multiplication. For example, 

2⁴ = 2×2 x 2 x 2 = 16. 

Here, exponent 2 is a whole number and is also called the base. In the number, say x^(1/y), x is the base, and 1/y is the fractional exponent.

A fractional exponent is a good way to represent powers and roots together. In any general exponent of the form a^b, if b is given in the fractional form, it is known as a fractional exponent. A few examples of fractional exponents are 3^(½), 3^(⅔), etc. Thus the general form of a fractional exponent is a^(m/n), where a is the base and m/n is the exponent.

Frequently Asked Questions

1. What does exponent stand for?

It stands for a number that tells you how many times to multiply something by itself.

2. What is an example of an exponent?

An exponent is a number that tells you how many times to multiply the number by itself. For example, 5^2 means 5 times itself 2 times: 555.

3. What does an exponent look like?

An exponent looks like a number with a little superscripted “x” above it.

For example:

42 = 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4

This is what the exponent looks like 42

4. How do you calculate exponents?

To calculate exponents, there’s a really simple formula: (base)^(power). So if we wanted to calculate what 25^3 is in our previous example, it would look like this: (25)^(3).

5. How to multiply with exponents?

You can multiply with exponents by adding the exponents.

Let’s look at an example: 2^1 + 2^2 = 2^3. So if you have two numbers, x and y, and their exponents are a and b respectively, then you can multiply them together by adding their exponents like this: ax + by = c.