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Multiplying Exponents – Rules, Definition With Examples

When talking about multiplying exponents, a lot of us, children and adults, would think it’s something difficult. Here in Brighterly, we show that difficult things can be explained in easy words. Today’s article will break the how to multiply exponents with different bases and same bases, what happens when you multiply exponents, share useful worksheets, and practice problems to consolidate the result. 

What are exponents?

Exponents represent repeated multiplication of the same number. The number being multiplied is called the base, and the exponent shows how many times the base is used as a factor. For instance, 3⁴ is equivalent to 3 × 3 × 3 × 3. Exponents help write large or repeated calculations in a shorter, clearer form. They are widely used in arithmetic and algebra to simplify expressions, compare values, and solve mathematical problems more efficiently.

Multiplying exponents – basic rule

When you multiply exponents expressions that have the same base, the base remains unchanged and the exponents are added together. This rule is called the Product Rule of exponents. It is written as:

a^m x aⁿ = a^m+n

So, when multiplying exponents do you add them? Yes!

Adding exponents works because you are combining repeated multiplication of the same base into a single power.

Multiplying exponents with same base

If the terms have the same base, keep it the same and add the exponents together. In exponents, this property is known as the Product Rule. The general formula is:

a^mx aⁿ = a^m+n

Let’s try to solve some examples.

Example 1: ^36x 3⁴= ?

The basis in this case is the same, which is 3. We shall add the powers in accordance with the rule. Let’s see what we get:

3⁶x 3⁴ = 3⁶⁺⁴= 3¹⁰

Example 2: 8² x  8³ = ?

In the given example, the bases are the same, that is, 8. Applying the rule, and adding the powers:

8² x  8³ = 8²⁺³= 8⁵

Multiplying exponents with different bases

By using some fundamental exponent laws, we can multiply expressions when two variables or integers have different bases. Here, we have the two circumstances listed below.

When the powers are the same but the bases are different

Examine two expressions, aⁿ and bⁿ, that have the same power but a different base. In this case, the power is n and the bases are a and b. So, when do you multiply exponents like this, what should we know? The bases are multiplied first when exponents with different bases and the same powers are multiplied. 

Mathematically, it can be expressed as:

a^m x b^m = (a x b)^m

Let’s try to solve several examples.

Example 1. 3³x 8³ = ?

In this case, the bases are different, but the powers are the same. Taking the rule into account, the first thing we would do is multiply the bases, which would be as follows:

3³ x 8³ = (3 x 8 )³ = 24³

Example 2. 43 x 9³ = ?

You can use the exponent on the product of the bases to combine exponents with distinct bases but the same exponent.

4³ x 9³ = (4 x 9)³ = 36³

When the powers and bases are different

Here are two expressions with different bases and powers, an and bm. Here, a and b are the bases. There are two powers, n and m. When multiplying expressions with different bases and powers, each expression is evaluated separately and then multiplied. Mathematically, it can be expressed as follows:

aⁿ x b^m = (aⁿ) x (b^m)

Example 1: 6² x 5³ = ?

Solution: We see the different bases and the powers here. That’s why, we need to solve each term separately.  6² x 5³ = 36 x 125 = 4500

Multiplying negative exponents

Negative exponents indicate how many times the base’s reciprocal must be multiplied. To put it another way, by expressing the reciprocal of the provided term and solving it as a positive term, we can change a negative exponent to a positive one. 

The general rule is:

a^-n=1/aⁿ

For instance, 4^-2 can be expressed as ^1/42. The following table outlines the rules for multiplying exponents with negative sign.

Let’s try to solve some examples. How do you multiply exponents in following examples?

Example 1: 4^-3x 4^-2= ?

4^-3x 4^-2=1/4³x 1/4²=1/4⁵

Example 2: 7^-2 x 7^-5 = ?

7^-2 x  7^-5 = 1/7² x 1/7⁵ = 1/7⁷

Multiplying exponents with variables

If the base of a term is a variable, we follow the same exponent rules of multiplication that are used for numbers. When the variable bases are the same, the powers are added.

• Powers are added when the variable bases are the same

Example: b³ x b⁵ = ?

It is the same variable base, ‘b’. Therefore, b³x b⁵ = b³⁺⁵ = b⁸ due to the exponents.

• Different variable bases and the same power are multiplied first if the bases are different

Example: Multiply 8⁸ x 7² = ?

Solution: The variable bases are different and the powers are the same, that is, 

8⁸ x 7⁸ =(8 x 7)⁸ = 56⁸

• Variable bases and powers are evaluated separately and multiplied when they differ

Example: a⁷ x b³= ?

Solution: Bases and powers of the variables are different, that is, a⁷ x b³ = a⁷b³

Multiplying exponents with fractions

Roots and powers are represented by fractional exponents. Multiplying bases with fractional exponents still follows the basic exponent rules. So, multiplying numbers with exponents with fractions, following the rule we have this formula:

a^1/2 x a^2/3 = a^{1/2 + 2/3}

Note: n√a = a1/n.

Example: √2 × √8.

√2 × √8 = 21/2 × 81/2 = (2 × 8)1/2 = 161/2 = 4

If we have different bases but the same fractional powers we’re multiplying like this:

c^n/mx y^n/m= (c x y)^n/m

For different bases and fractional powers you can use this formula:

a^m/n x b^c/y = (a^m/n) x (b^c/y) 

Example: Multiply 3^1/3 x 4^2/5 = ?

Solution: In this example, we have different bases and fractional powers So, let’s rewrite each term:

3^1/2= 3

4^2/5= (2²)^2/5 = 2^4/5

Since the bases are different they cannot be combined, so the final answer is: 2^4/5√3

Difference between adding and multiplying exponents

Adding exponents and multiplying exponents follow different rules. To answer the question When you multiply exponents do you add them we need to know basic rules. When adding exponents, the bases must be the same and the expressions are usually being multiplied. When multiplying exponents, you are finding the product of powers, not combining their values. For example, a² + a³ cannot be simplified, but a² x a³ =a⁵ using the Product Rule.

Solved examples on multiplying exponents

Example 1. Multiply 4²x 4⁵ =?

Solution: Since the bases are the same, add the exponents. Using the same base rule:

4²x 4⁵ = 4²⁺⁵ = 4⁷

Example 2: Simplify (x⁴)³ = ?

Solution: Use the power of a power rule:

(x4)³ = x^4x3 = x¹² 

Example 3: How to multiply exponents 3^-2 x 3^-4 = ?

Solution: Add the exponents. A negative exponent means the reciprocal.

3^-2 x 3^-4 = 1/3² x 1/3⁴ = 1/3⁶

Example 4: Simplify 7^1/3 x 7^2/3 = ?

Solution: With the same base, add the fractional exponents:

7^1/3 x 7^2/3 = 7^{(1/3 + 2/3)} = 7¹ = 7

Practice problems on multiplying exponents

1. Multiply b³ x b^-4 = ?
   1. b^-7
   2. b¹²
   3. b-¹²
   4. b^-1 — true

2. Find the right answer 2² x 2⁴ = ?
   1. 2⁶ — true
   2. 2⁴
   3. 2⁸
   4. 2²

3. How to multiply numbers with exponents 2^-3x 2^-2 = ?
   1. 2⁶
   2. 2^-6
   3. 1/2⁶
   4. 1/2⁵ — true

4. Simplify 3^1/2 x 3^3/2 = ?
   1. 3²
   2. 9 — true
   3. 3³
   4. 3^1/3

Frequently Asked Questions on Multiplying Exponents

Do you add exponents when multiplying?

Yes, you need to add exponents when multiplying terms with the same base. When the term stays the same, and the powers are added together, we follow the  main exponent multiplication rules — Product Rule: aᵐ × aⁿ = aᵐ⁺ⁿ.

What to do when multiplying exponents?

When multiplying exponents, first, check if the bases and exponents are the same. If bases are the same, add exponents. If exponents are the same, multiply the bases. Otherwise, evaluate each term separately before multiplying with exponents.

What is the rule for multiplying exponents?

The main rule states that when multiplying powers with the same base, keep the base unchanged and add the exponents. For different bases with the same exponent, multiply the bases and keep the exponent.

Do you add or subtract exponents when multiplying?

When multiplying exponents, you add the powers only if the bases are the same. Subtraction of exponents is used in division, not multiplication, according to the exponent multiplication rules explained in the article.

Multiplying exponents worksheets

We also have prepared multiply exponents worksheets to help your kids practice. They include tasks with the same and different bases, negative and fractional exponents, for different:. 

• Multiplying exponents worksheets
• Multiplying exponents with same base worksheet

With Brighterly, you will know when to multiply exponents and how to do it more easily. By learning multiplying exponents rules, you can solve complex expressions with no fear.