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.

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 a^m ÷ aⁿ, where 'a' is the common base and 'm' and 'n' are the exponents. This 'Quotient property of Exponents' says, a^m ÷ aⁿ = a^m-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, 6^{5 - 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 = a^m/b^m. Consider a^m ÷ b^m, 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 = a^{7 - 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, a^m × 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: a^m × b^m = (a × b)^m. For example, let us multiply: 11⁴ × 3⁴. This can be solved as, 11⁴ × 3⁴ = (11 × 3)⁴ = 33⁴.

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, 2^½, 6^¾, and so on.

Multiplying Fractional Exponents with the Same Base

In order to multiply fractional exponents with the same base, we use the rule, a^m × aⁿ = a^m+n. For example, let us simplify, 2^½ × 2^¾ = 2^{(½ + ¾ )} = 2^5/4.

Dividing Fractional Exponents with the Same Base

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

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:

• a^m × aⁿ = a^m+n
• a^m × b^m = (a × b)^m
• a^m ÷ aⁿ = a^m-n
• a^m ÷ b^m = (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: a^m × b^m = (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: a^m ÷ aⁿ = a^m-n , we get 5^{(2x -1 - x - 1)} = 5^{(x -2)}

Tips on Multiplying and Dividing Exponents

• a⁰ = 1[ since a^m ÷ a^m = 1 = a^m-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

How do you Multiply and Divide Exponents?

In order to multiply and divide exponents, we use a set of exponent rules. When two exponential terms with the same base are multiplied, their powers are added while the base remains the same. For example, let us multiply, 6³ × 6⁵ = 6^{(3 + 5)} = 6⁸. However, when two exponential terms having the same base are divided, their powers are subtracted. For example, 7⁸ ÷ 7⁵ = 7³. Similarly, there are other rules that help to simplify exponents easily.

What are the Rules for Dividing Exponents?

There are a few exponent rules that help in the division of exponents. These rules also help in simplifying numbers with complex powers involving fractions, decimals, and roots. For example, in order to divide the numbers or variables with the same base, we apply the rule: a^m ÷ aⁿ = a^m-n. To divide the numbers or variables with different bases, we apply the rule: a^m ÷ b^m = (a ÷ b)^m

How do you Solve Exponents in Parentheses?

The exponents inside parentheses can be solved using the identity (a^m) ⁿ = a^mn. For example, (4²)³ = 4^{(2 × 3)} = 4⁶ = 4096

Can we Distribute Exponents Over Division?

Yes, we can distribute exponents over division. For example, (7/2) ³ = 7³ ÷ 2³ = 343/8

How to Multiply and Divide Negative Exponents?

When we multiply and divide negative exponents, we follow the same rules that are used for positive exponents. For example, we use the property: a^m ÷ aⁿ = a^m-n, to solve: 2^-3 ÷ 2^-4. This will be: 2^(-3-(-4)) = 2^{(-3 + 4)} = 2¹ = 2. Observe the rule of simplification of integers which changes the sign after the brackets are opened. 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, 7^-3 can also be written as: 1/7³. This means that if we need to divide expressions that have negative exponents, we can simply move the base to the other side of the fraction bar. For example, if we have 4^-2 in the denominator of a fraction, we can move it to the numerator. This means, y^-2/y^-3 = y³/y² = y^{3 - 2} = y¹ = y.

How to Divide Exponents with Different Powers?

In order to divide exponents with different powers, but the same bases, we subtract the given powers. The property which is used here is, a^m ÷ aⁿ = a^(m-n). For example, let us divide the exponents, 8⁶ ÷ 8⁴. After applying the Quotient property of exponents, we get, 8^{6 - 4} = 8²

How to Divide Exponents with Fractions?

In order to divide exponents with fractions, we use the same rule that is used for whole numbers, that is, a^m ÷ aⁿ = a^m-n. For example, let us divide the following exponents, 2^3/4 ÷ 2^1/2 = 2^{3/4 -1/2} = 2^1/4.

How to Divide Exponents with Different Bases and Same Powers?

In order to divide exponents with different bases and the same powers, we apply the 'Power of Quotient Property' which is, a^m ÷ b^m = (a ÷ b)^m. For example, let us divide, 14³ ÷ 2³ = (14 ÷ 2)³ = 7³.

How to Divide Exponents with Negative Bases?

When we need to divide exponents with negative bases, the exponent rules remain the same. For example, let us divide (-4)⁸ ÷ (-4)² = (-4)^{8 - 2} = (-4)⁶