Adding Exponents

Adding exponents refers to the simple addition of numbers but in the form of exponents or power. In other words, the addition of base and exponents is adding exponents. An exponent is also called the power of a number and it shows how many times the number is multiplied by itself. In general, xⁿ means that x is multiplied by itself for n times. Adding exponents is done in different types, let us see what those types are and solve a few examples to understand this concept better.

Adding exponents is the process of adding exponents or powers of a number irrespective of the base being the same or not. Exponents can also be called the power of the numbers as it represents the number of times a number is multiplied by itself. For example, 3² = 3 × 3, where 3 is the base and 2 is the exponent.

Here, in the term xⁿ,

• x is called the base
• n is called the exponent or power
• xⁿ is read as x to the power of n (or) x raised to n

While adding exponents, the one main rule to be remembered is the base and exponent need to be the same and the addition is performed on the coefficient. The variables that combine have the same base and the same power. This rule is applicable in all the other forms of exponents as well i.e subtraction, multiplication, and division.

Adding exponents can be performed when the base and exponents are the same. There would be times when the base and exponents are different but we can still perform adding for those expressions. Let us look at the steps of adding exponents.

• Step 1: Check the terms in the expression if they have the same base and same exponents. For example, 2² + 2². As we can see, both the base and exponent are 2.
• Step 2: If the base and exponents are different, calculate the expression with individual terms. For example, 5³ + 4². The base and exponents are different.
• Step 3: Add the results together.

Addition of exponents can be performed in different methods. The basic rule while adding exponents is that the base and exponents need to be the same. However, sometimes the base and exponents might not the same, so we need to calculate the terms individually to calculate the expression. Let us see the different methods.

Method 1: Adding Exponents With Same Base and Exponents

Adding exponents when the base and exponents are the same is done in a very simple method. The general form of adding exponents with the same base and exponents is aⁿ + aⁿ = 2aⁿ. Let us look at example to understand this better. For example: 4³ + 4³ = 2(4³) = 2 × 4 × 4 × 4 = 128.

Method 2: Adding Exponents With Different Base and Exponents

When the base and exponent are of different values, we first add each exponent first and then calculate the entire expression. The general form of calculating different bases and exponents is aⁿ + b^m. Let us look at an example to understand this better. For example: 3³ + 5² = 3 × 3 × 3 + 5 × 5 = 27 + 25 = 52.

Method 3: Adding Negative Exponents With Different Bases

Adding negative exponents is done by calculating each term individually and then add the total. The terms are written in a fractional form and then added. The general form of calculating negative exponents with different bases is a^-n + b^-m = 1/aⁿ + 1/b^m. Let us apply the general form in an example to understand this better. For example: 6^-2 + 3^-3 = 1/6² + 1/3³ = 1/36 + 1/27 = 0.0648.

Method 4: Adding Fractional Exponents With Same Base and Exponents

Exponents can be expressed in the form of a fraction as well. So adding these fractions in the form of exponents can be done using this general form, a^n/m + a^n/m = 2a^n/m. For fractions, we can convert the fractional exponents in the form of root i.e. either square root or cube root depending on the fraction. Let us apply this general form in an example for a better understanding. For example: 4^1/2 + 4^1/2 = 2(4^1/2) = 2 × √4 = 2 × 2 = 4.

Method 5: Adding Fractional Exponents With Different Base and Exponents

Exponents that are written in the fractional form with different base and different exponents is expressed in the general form as a^n/m + b^d/c. Here, each term is calculated first and then the whole result is calculated. The fractional form is converted into its root and then calculated. Let us look at an example, 27^1/3 + 4^1/2 = ³√27 + √4 = 3 + 2 = 5.

Method 6: Adding Variable With Same Exponents

This is similar to the method of adding exponents with the same base and same exponents. The general form is xⁿ + xⁿ = 2xⁿ. Let us look at example, 7² + 7² = 2(7²) = 2 × 7 × 7 = 98.

Method 7: Adding Variable With Different Exponents

The general form to calculate variables with different exponents is xⁿ + x^m. Let us look at an example, 4² + 4³ = 4^{2 × 3} = 4⁶ = 4096.

Related Topics

Listed below are a few interesting topics related to adding exponents. Take a look.

• How to Identify Exponents and Powers
• Exponential Functions
• Irrational Exponents

What is Meant by Adding Exponents?

Adding exponents is the process of the addition of terms that have the same base and the exponent. Adding can only happen if the base and exponent are the same. There are cases when they are not, it can either be solved by seeing if the exponents of two terms are the same or the base of two terms is the same. If not, adding in exponents cannot take place.

What are the Steps to Adding Exponents?

Adding exponents is done in 3 simple steps, they are:

• Check the terms in the expression if they have the same base and same exponents.
• If the base and exponents are different, calculate the expression with individual terms.
• Add the results together.

What is the Rule of Adding Exponents?

The most important rule of adding exponents is that the base and the exponents of the terms that are being placed for addition have to be the same. If they are the same, the coefficients will be added together, while the base and exponent is the same.

Can You Add Numbers With Different Exponents?

No, adding numbers with different exponents is applicable as the rule of exponent addition is that the base and exponent should be the same. While something the base and exponent might be different but just one being different is not applicable.

Do You Add Exponents While Adding?

For adding exponents, the base and the exponent should be the same. The coefficient of the variable is added leaving the exponent unchanged. In the expression, the terms with the same variables and powers are added. This rule applies for both multiplication and division as well.

How Do You Add Monomials With Exponents?

Adding two or more like monomials, we first add the coefficients while the variables and exponents of the variable are the same. We can obtain the result by adding the like monomials.