Rules for Multiplying Exponents

In order to understand how to multiply exponents, it is important to understand what exponents are. In the simplest of terms, an exponent is a shorthand way of notation for when a number is being repeatedly multiplied with itself.

Therefore, if you are multiplying the number 2 thrice, i.e., 2 × 2 × 2, then you could notate it as 2 raised to the power 3 or 2³. The rules for multiplying base numbers with exponents is slightly different from normal multiplication and here we try to explain those rules.

In order to understand how to multiply exponents, it is important to understand the different properties of exponents. Without a proper understanding of the base and exponent rule, it is impossible to solve any problem that you are faced with.

One basic rule that you need to always remember is that any base number raised to the power 0 is always equal to the number 1. Here are all the important rules that you need to know and understand if you want to multiply exponents.

This is the rule for multiplying two exponents or powers that have the same base number. If this is the case, then all you need to do is keep the base number common and add the two exponents. The resultant exponent will be the new power for the base number. The exponential notation for this rule is:

x^a × x^b = x^{(a + b)}

To understand this rule better it is important to take a look at an example for the same which is given below:

2² × 2³ = 2^{(2 + 3)} = 2⁵= 32

Multiplying exponents when they have the same base is extremely simple as all you need to do is add the exponent powers together.

Multiplication rules for exponents of different base numbers is such that there is a shorter method of multiplication only if the exponent numbers are the same.

If the base number and the exponent number are both different, then to multiply the two numbers, you will first need to simplify the base numbers and then multiply the two. For multiplying two different base numbers with the same exponent the rule is detailed below.

x^a × y^a = (x × y)^a

(xy)^a = x^a × y^a

To understand this rule better let us take a look at the example given below that should help you understand how the process of multiplication takes place.

2² × 3² = (2 × 3)² = 6²= 36

The process of multiplying different base numbers with the same exponential power is as easy as this.

Often you may be faced with a multiplication problem where the base number will have two exponent powers. In order to solve the power of a power, you will need to multiply the exponents together and then this would be the new exponent for the base number. This explanation will be made clear with the notational rule given below.

x^(a)b = x ^{(a × b)} = x^(ab)

The rule for multiplying an exponent which is the power for another exponent can be understood better with the example given below.

2⁽²⁾³ = 2 ^{(2 × 3)} = 2⁽⁶⁾ = 64

These are the three basic rules that you will need to follow when you are multiplying exponents. The division of exponent numbers is also irrevocably linked to multiplication. The rule for division of exponents is given below.

x^a ÷ x^b = x^(a)(1/x^b) = x^{(a+ (-b))} = x^{(a -b)}

If this is too confusing for you, then refer to the example given below for the same.

2⁵ ÷ 2³ = 2⁽⁵⁾(1/2³) = 2^{(5+ (-3))} = x^{(5 -3)} = 2² = 4

While solving algebra problems, you need to be not only familiar but also proficient with all the different rules that you need to keep in mind for multiplying exponents. Hope you understand how to solve a multiplication problem that involves exponents.