Adding Exponents

Algebra forms one of the core areas of mathematics. Algebra is broadly categorized into various fields, ranging from elementary algebra that we learn during our school years to higher-order algebra like linear algebra, abstract algebra and vector algebra, etc. Exponents are a part of algebra syllabus and a command in the basic concepts of exponents is essential for kids and students to have strong base in mathematics. Adding exponents is often perceived by students to be similar to addition of numbers and hence they tend to make many mistakes. To understand algebra, learning proper usage of exponents and radicals is essential.

How to Add Exponents?

Exponents, in simplest terms, are repeated multiplication of the same thing by itself. For example, if you have to multiply 3 four times, you can represent it mathematically as (3 x 3 x 3 x 3) or 3⁴. The exponent in this example is 4 and the base to which the exponent is raised is 3. So did you get what exactly is exponentiation? Well, exponentiation in mathematics is defined as a mathematical operation involving numbers in the form a^m on which all the basic operations like addition, subtraction, division and multiplication hold true.

The common problem encountered by students while adding exponents is that they are not able to understand how to add exponents when the bases are same/different or exponents are same/different or both are different.

Exponent Addition With Same Bases
Consider an example with like bases: 4³ + 4⁵. In this example, the base of both the exponents is same, 4. However, their exponents or powers are different. How to add them? Generally, students make a common mistake in solving this problem by adding both the exponents and raising it the power of same base,

4³ + 4⁵ = 4^{(3 + 5)} = 4⁸ (Wrong Solution)

The above solution is absolutely wrong because if bases are same, you just can't add the exponents.

Let us see the correct solution. We know that 4³ = 64 and 4⁵ = 1024. So 4³ + 4⁵ = 64 + 1024 = 1088. And as per our previous solution, 4³ + 4⁵ = 4⁸ = 65, 336 that contradicts the result 1088!!

Another simple method for adding exponents with same base is as follows.
4³ + 4⁵ = ?
Step 1: 4³ (1 + 4²) (Taken 4³ common)
Step 2: 4³ (1 + 16)
Step 3: 64 x 17 = 1088 (Answer)

This approach is easier and eases the calculations.

So the general rule is: a^m + aⁿ ≠ a^{m + n}

Exponent Addition With Different Bases
Let us consider an example where there are different bases. For instance, consider adding these two, 3⁴ + 5² = ? As evident from the question, there is nothing common in both the numbers, so going by logic, all we have to do is to just find the sum of the numbers 3⁴ = 81 and 5² = 25, that is 81 + 25 = 106.
Therefore, 3⁴ + 5² = 106.

Likewise, adding exponents with variables is also possible and it follows the same rules of exponents. Here are some more basic rules of exponents.

• aⁿa^m = a^n+m
• (a.b)ⁿ = aⁿ. bⁿ
• a⁰ = 1
• (a^m)ⁿ = a^mn
• a^m/n = n√a^m
• a^-m = 1/a^m
• (a^m/aⁿ) = a^(m-n)

Indeed, solving exponents is not a difficult task once we understand the basics of this mathematical operation. With regular practice, rules of exponents can be easily memorized and once students are able to grasp the rules, they will have fun while solving mathematical problems related to exponents.