When To Multiply Or Add Exponents (3 Key Concepts)

Multiplying and adding exponents is often used in algebra to simplify expressions and to solve equations.  However, you need to know when to use each operation so you get the right answer every time.

So, when do you multiply and add exponents?  We multiply exponents when we have a base raised to a power in parentheses that is raised to another power. For example, (2³)⁴ = 2^3*4 = 2¹². We add exponents when we have a product of two terms with the same base. For example, 2³*2⁴ = 2³⁺⁴ = 2⁷.

Of course, there are other special cases to be aware of.  For example, when we divide two terms with the same base, we subtract the exponents: 2⁷ / 2⁴ = 2^7-4 = 2³.

In this article, we’ll talk about when to multiply and add exponents.  We’ll also answer some common questions and look at some examples to make the concepts clear.

Let’s get started.

When To Multiply & Add Exponents

When we multiply exponents, it is really a special case of adding exponents.  We can prove this with the general formulas for adding and multiplying exponents (more on this later).

Let’s start off with when to add exponents, along with some examples of how it works.

When To Add Exponents

The first thing to remember is that we add exponents when we multiply two terms with the same base.

The general equation for adding exponents is given by the formula:

• x^A*x^B = x^{A + B}  [where x is the common base]

Example 1: Adding Exponents

Consider the product 2*2.  Since both terms have the same base (here, the base is 2), we add the exponents.

Remember that when there is no exponent written, it is assumed to be 1:

• 2*2
• = 2¹*2¹  [assume an exponent of 1 if none is written]
• = 2¹⁺¹  [add exponents: A = 1, B = 1]
• = 2²

We can also add exponents when the exponents are not 1, and when they are different.

Example 2: Adding Exponents

Consider the product 2³*2⁵.  Since both terms have the same base (here, the base is 2), we add the exponents.

• 2³*2⁵
• = 2³⁺⁵  [add exponents: A = 3, B = 5]
• = 2⁸

We can also add exponents when the base is a variable.

Example 3: Adding Exponents When The Base Is A Variable

Consider the product x²*x⁷.  Since both terms have the same base (here, the base is the variable x), we add the exponents.

• x²*x⁷
• = x²⁺⁷  [add exponents: A = 3, B = 5]
• = x⁹

We can also add exponents when the exponents are variables.

Example 4: Adding Exponents When the Exponents Are Variables

Consider the product 3^4x*3^2x.  Since both terms have the same base (here, the base is 3), we add the exponents.

• 3^4x*3^2x
• = 3^4x+2x  [add exponents: A = 4x, B = 2x]
• = 3^6x

Can You Add Exponents With Different Bases?

You cannot add exponents with different bases.  To add exponents, the two terms in the product must have the same base.

However, you can sometimes change the base by using powers.  You can also use logarithms to change the base on one or both terms to get the same base.

Then, you could add the exponents as usual.

Example 1: Adding Exponents After A Change Of Base

Consider the product 3⁵*9³.  Since the terms have different bases (here, 3 and 9), we cannot add the exponents.

However, we can change the base on the 2^nd term to make the bases the same.  We will use the fact that 9 = 3²:

• 3⁵*9³
• = 3⁵*(3²)³  [since 9 = 3²]
• = 3⁵*(3^2*3)  [since (3²)³ = 3^2*3]
• = 3⁵*3⁶  [now we can add exponents, since the base is 3 for both terms in the product]
• = 3^{5 + 6} 
• = 3¹¹

Sometimes, we may need to use logarithms to make a change of base, but the idea is the same.

Example 2: Adding Exponents After A Change Of Base With Logarithms

Consider the product 2⁴*5.  Since the terms have different bases (here, 2 and 5), we cannot add the exponents.

However, we can change the base on the 2^nd term to make the bases the same.  We will use the fact that 5 = 2^log_2(5):

• 2⁴*5
• = 2⁴*2^log_2(5)  [5 = 2^log_2(5)]
• = 2^{4 + log_2(5)}    [now we can add exponents, since the base is 3 for both terms in the product]

Can You Add Fractional Exponents?

You can add fractional exponents in the same way as you do for whole number exponents – you just need to find a common denominator.

The formula is below:

• (a / b) + (c / d) = (ad + bc) / bd  [formula for adding fractions; this does not guarantee that the common denominator bd is the least common denominator, or LCD]

Example 1: Adding Fractional Exponents

Consider the product 2^1/2*2^3/5.  Since both terms have the same base (here, the base is 2), we add the exponents.

• 2^1/2*2^3/5
• = 2^{(1/2) + (3/5)}  [add exponents: A = 1/2, B = 3/5]
• = 2^{(5/10) + 6/10)}  [use a common denominator of 10 = 2*5]
• = 2^{(5 + 6)/10}  [add numerators and use the common denominator]
• = 2^11/10

Do You Add Exponents When Dividing?

We subtract exponents when we divide two terms with the same base.  To be precise, we add the negative of the exponent in the denominator term when we divide two terms with the same base.

Here is the formula:

• x^A / x^B = x^{A – B}  [where x is the common base]

This formula comes from the rule for adding exponents that we mentioned earlier.  Let’s see the proof:

• x^A / x^B
• = x^A * (1 / x^B)
• = x^A * (x^-B)  [since 1 / x^B = x^-B]
• = x^{A + -B}  [since we add exponents when we multiply terms with the same base]
• = x^{A – B}

Technically yes; negate exponent; equivalent to subtracting – subtract them (top minus bottom);.

Example 1: Subtracting Exponents

Consider the quotient 7⁹ / 7⁵.  Since both terms have the same base (here, the base is 7), we subtract the exponents.

• 7⁹ / 7⁵
• = 7^{9 – 5}  [subtract exponents: A = 9, B = 5]
• = 7⁴

We can also subtract exponents when the base is a variable.

Example 2: Subtracting Exponents When The Base Is A Variable

Consider the quotient x⁸ / x⁶.  Since both terms have the same base (here, the base is the variable x), we subtract the exponents.

• x⁸ / x⁶
• = x^{8 – 6}  [subtract exponents: A = 8, B = 6]
• = x²

We can also subtract exponents when the exponents are variables.

Example 3: Subtracting Exponents When The Exponents Are Variables

Consider the quotient 3^11x / 3^4x.  Since both terms have the same base (here, the base is 3), we subtract the exponents.

• 3^11x / 3^4x
• = 3^{11x – 4x}  [subtract exponents: A = 11x, B = 4x]
• = 3^7x

Do You Add Exponents When Adding Or Subtracting?

We do not add exponents when adding or subtracting two terms.

For example, let’s say we want to add 3² and 3⁴.

We get a result of 3² + 3⁴ = 9 + 81 = 90.

If we add the exponents, we get the wrong answer:

3²⁺⁴ = 3⁶ = 729.

When To Multiply Exponents

The first thing to remember is that we multiply exponents when we have a base to a power in parentheses, raised to another power.

The general equation for multiplying exponents is given by the formula:

• (x^A)^B = x^A*B  [where x is the common base]

Remember that this comes from the rule for adding exponents when the base is the same, since:

• (x^A)^B
• = x^A * x^A * … * x^A  [we multiply x^A, B times]
• =x^{A + A + … + A}  [we add the exponent A, B times]
• = x^AB  [since A + A + … + A, B times, is AB]

Now let’s look at some examples of how this rule is applied.

Example 1: Multiplying Exponents

Consider the expression (2³)⁵.  Since we have a base raised to a power in parentheses that is raised to another power, we multiply exponents.

• (2³)⁵
• = 2^3*5  [the exponents are A = 3 and B = 5]
• = 2¹⁵  [multiply the exponents]

We can also multiply exponents when the base is a variable.

Example 2: Multiplying Exponents When The Base Is A Variable

Consider the expression (x⁴)⁶.  Since we have a base raised to a power in parentheses that is raised to another power, we multiply exponents.

• (x⁴)⁶
• = x^4*6  [the exponents are A = 4 and B = 6]
• = x²⁴  [multiply the exponents]

We can also multiply exponents when the exponents are variables.

Example 3: Multiplying Exponents That Are Variables

Consider the expression (2^3x)^4y.  Since we have a base raised to a power in parentheses that is raised to another power, we multiply exponents.

• (2^3x)^4y
• = 2^3x*4y  [the exponents are A = 3x and B = 4y]
• = 2^12xy  [multiply the exponents]

Can You Multiply Negative Exponents?

You can multiply negative exponents.  We apply the usual rules for signs: a product of two negatives gives us a positive.

Example 1: Multiplying Negative Exponents

Consider the expression (5³)^-2.  Since we have a base raised to a power in parentheses that is raised to another power, we multiply exponents.

• (5³)^-2
• = 5^3*(-2) [the exponents are A = 3 and B = -2]
• = 5^-6  [multiply the exponents]
• =1 / 5⁶  [by definition of negative exponents]

Example 2: Multiplying Negative Exponents

Consider the expression (7^-4)^-3.  Since we have a base raised to a power in parentheses that is raised to another power, we multiply exponents.

• (7^-4)^-3
• = 7^(-4)*(-3) [the exponents are A = -4 and B = -3]
• = 7¹²  [multiply the exponents; a product of two negatives is positive]

Can You Multiply Fractional Exponents?

You can multiply fractional exponents in the same way as you do for whole number exponents.

Remember that to multiply two fractions, we multiply numerators to get the new numerator, and multiply denominators to get the new denominator.

The formula is below:

• (a / b)*(c / d) = ac / bd  [formula for multiplying fractions]

Example 1: Multiplying Fractional Exponents

Consider the expression (7⁵)^1/3.  Since we have a base raised to a power in parentheses that is raised to another power, we multiply exponents.

• (7⁵)^1/3
• = 7^5*(1/3) [the exponents are A = 5 and B = 1/3]
• = 7^5/3  [multiply the exponents; treat 5 as 5 / 1]

Example 2: Multiplying Fractional Exponents

Consider the expression (6^2/3)^4/5.  Since we have a base raised to a power in parentheses that is raised to another power, we multiply exponents.

• (6^2/3)^4/5
• = 7^(2/3)*(4/5) [the exponents are A = 2/3 and B = 4/5]
• = 7^(2*4/3*5)
• = 7^8/15

Conclusion

Now you know when to multiply exponents and when to add exponents.  You also know when you will need to subtract exponents.

I hope you found this article helpful.  If so, please share it with someone who can use the information.

Don’t forget to subscribe to my YouTube channel & get updates on new math videos!

~Jonathon