Multiplying And Dividing Exponents - Rules, Examples - Cuemath

Multiplying and Dividing Exponents

An exponent shows how many times a given variable or number is multiplied by itself. For example, 64 means we are multiplying 6 four times. In the expanded form, it is written as 6 × 6 × 6 × 6. When two exponential terms with the same base are multiplied, their powers are added while the base remains the same. However, when two exponential terms having the same base are divided, their powers are subtracted. Let us learn more about multiplying and dividing exponents in this article.

1.	Dividing Exponents
2.	How to Multiply and Divide Fractional Exponents?
3.	How to Multiply and Divide Exponents with Variables?
4.	FAQs on Multiplying and Dividing Exponents

Dividing Exponents

The laws of exponents make the process of simplifying expressions easier. The basic rule for dividing exponents with the same base is that we subtract the given powers. This is also known as the Quotient Property of Exponents.

How to Divide Exponents?

Dividing exponents becomes easy when we follow the properties of exponents. For example, let us solve the following question in the usual way, 65 ÷ 63 = (6 × 6 × 6 × 6 × 6)/(6 × 6 × 6 ) = 62. This involves more calculation. However, when we use the laws of exponents, it reduces all these calculations. Let us understand how to divide exponents in different scenarios using the different properties.

Dividing Exponents with Same Base

In order to divide exponents with the same base, we use the basic rule of subtracting the powers. Consider am ÷ an, where 'a' is the common base and 'm' and 'n' are the exponents. This 'Quotient property of Exponents' says, am ÷ an = am-n. Now, let us understand this with an example.

Example: Divide 65 ÷ 63

Solution: We can see that in the given expression, the bases are the same. Using the 'Quotient property of Exponents', we will get, 65 - 3 = 62. Therefore the answer is 62.

Dividing Exponents with Different Bases

In order to divide exponents with different bases and the same exponent, we use the 'Power of quotient property', which is, (a/b)m = am/bm. Consider am ÷ bm, where the expressions have different bases and the same exponent. For example, let us solve: 123 ÷ 33. Using the 'Power of quotient property', this can be solved as, 123 ÷ 33 = (12 ÷ 3)3 = 43

Dividing Exponents with Coefficients

In some cases, we need to divide expressions that have coefficients. These coefficients that are attached to their bases can be divided easily in the same way as we divide any other fraction. It should be noted that the coefficients can be divided even if the expressions have different bases.

Example: Divide 12a7 ÷ 4a2

Solution: Let us use the following steps to divide expressions with coefficients. In this case, 12 and 4 are the coefficients and the rest are variables.

• First, we rewrite the expression as a fraction, that is, 12a7/ 4a2.
• Then we divide the coefficients, that is, 12/4 = 3.
• After this step, we can apply the quotient property of exponents and solve the variable, that is, a7/a2 = a7 - 2 = a5.
• So, now we have the coefficient as 3 and the variable is a5. This gives the answer as, 3a5

Multiplying Exponential Terms

Multiplying exponents with the same base and different bases involves certain rules of exponents. Let us understand these in the following section.

Multiplying Exponents with the Same Base

When we multiply two expressions with the same base, we apply the rule, am × an = a(m + n), in which 'a' is the common base and 'm' and 'n' are the exponents. For example, let us multiply 22 × 23. Using the rule, 22 × 23 = 2 (2 + 3) = 25.

Multiplying Exponents with Different Base and Same Power

When we multiply expressions with different bases and the same power, we apply the rule: am × bm = (a × b)m. For example, let us multiply: 114 × 34. This can be solved as, 114 × 34 = (11 × 3)4 = 334.

How to Multiply and Divide Fractional Exponents?

In order to multiply and divide fractional exponents, we use the same exponent rules that we apply for whole numbers. Fractional exponents are those expressions in which the powers are fractions, for example, 2½, 6¾, and so on.

Multiplying Fractional Exponents with the Same Base

In order to multiply fractional exponents with the same base, we use the rule, am × an = am+n. For example, let us simplify, 2½ × 2¾ = 2(½ + ¾ ) = 25/4.

Dividing Fractional Exponents with the Same Base

For dividing fractional exponents with the same base, we use the rule, am ÷ an = am-n. For example, let us solve, 33/2 ÷ 31/2. Using the rule, we get, 3(3/2 - 1/2) = 31 = 3.

How to Multiply and Divide Exponents With Variables?

The rules which are used in numbers are also used in exponents with variables. Let us recollect them and then use them in the following examples:

• am × an = am+n
• am × bm = (a × b)m
• am ÷ an = am-n
• am ÷ bm = (a ÷ b)m

Variable as the Base

Let us see how to use these rules when the base is a variable. For example, solve: y2 × (2y)3

We will apply the rule: am × bm = (a × b)m , y2 × (2y)3 = y2 × 23 × y3 = 23 × y(2+3) = 8y5

Variable as the Exponent

Let us see how to use the rules when the exponent is a variable. For example, solve: 5(2x -1) ÷ 5(x + 1)

We will apply the rule: am ÷ an = am-n , we get 5(2x -1 - x - 1) = 5(x -2)

Tips on Multiplying and Dividing Exponents

• a0 = 1[ since am ÷ am = 1 = am-m = a0]
• It should also be noted that a negative exponent can be converted to a positive exponent by writing the reciprocal of the number. For example, 6-3 can be written as 1/63.
• If we multiply two exponents with the same base then their powers will add.
• If we divide two exponents with the same base then their powers will subtract.

Related Topics

• Rational Exponents
• Irrational Exponents