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Adding Exponents

Adding Exponents

The rules for adding exponents are different from adding integers, whole, or fractional numbers. Here is some information about various rules to add exponents.
Kundan Pandey
Last Updated: Jun 3, 2018
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.

The confusion usually pertains to the difference in the meaning of exponents and exponentiation. Here is a detailed explanation of the meaning of these terms and some information about various rules to add exponents.
What are Exponents?
Exponents, in simplest terms, are repeated multiplication of the same thing by itself. Such operations in mathematics are known as exponentiation. It involves two numbers and is written as bn, where b is called the base and 'n' is known as the exponent or index or power.

Well, exponentiation is a mathematical operation involving numbers in the form ab, on which all the basic operations like addition, subtraction, division, and multiplication hold true.
Terminology
The index or exponent is always on the right side of the base as a superscript. It is generally read as:
  • a2 as a squared (derived from area of a square is l.l or l squared)
  • b3 as b cubed (derived from volume of cube is l.l.l or l cubed)
Therefore, 42 is "four squared" and 53 is "five cubed".
For all others, it is pronounced as "b raised to the power n-th power", or "b raised to the power of n" or simply "b raised to n".
Positive Exponents
Exponents with a positive integer as the base (non-zero) and index, are positive exponents. For example:
26, 345 and 6789
are all positive exponents. It is important to note here that x4 and 53, when expanded, will become (x)(x)(x)(x) and (5)(5)(5). Moreover, 6x2 will be
6(x)(x) and not (6x)(6x)
Examples:
  • a3 = (a)(a)(a)
  • 3a2 = 3(a)(a)
  • (3a)2 = (3a)(3a)
  • a2b = (a)(a)(b)
  • 2ab2 = 2a(b)(b)
  • 2a2b2 = 2(a)(a)(b)(b)
  • (2ab)2 = (2ab)(2ab)

So, in absence of parentheses, exponent or index will apply to the integer it is attached to. Always remember, exponent is first operation.

Many students face problems when it comes to adding positive exponents. And again, the cause of all the confusion is the term 'exponent'. So, to understand, simply remember:

xy is exponentiation where y is the exponent.

Now, when we say addition of exponents, it actually is multiplication of two exponentiations.
x(2 + 3) = x2. x3 = x5
x2 = (x)(x)
x3 = (x)(x)(x)
Now, for multiplying them together;
x2. x3 = (x)(x)(x)(x)(x)
Also,
63. 64 = 6(3 + 4) = 67 = 279936
Examples:
  • 52. 54 = 52 + 4 = 56 = 15625
    Check:
    52 = (5)(5) = 25
    54 = (5)(5)(5)(5) = 625
    25 * 625 = 15625
  • 73. 75 = 73 + 5 = 78 = 5764801
    Check:
    73 = (7)(7)(7) = 343
    75 = (7)(7)(7)(7)(7) = 16807
    343 * 16807 = 5764801

But this is true for exponents with same 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.

There exists an exception. In case the bases are different and the powers are same, you CAN combine them.
(a4)(b4)
= (a)(a)(a)(a)(b)(b)(b)(b)
= (ab)(ab)(ab)(ab)
= (ab)4
Examples:
  • (43)(53) = (4 * 5)3 = 203 = 8000
    Check
    (43) = 64
    (53) = 125
    64 * 125 = 8000
  • (102)(52) = (10 * 5)2 = 502 = 2500
    Check
    (102) = 100
    (52) = 25
    25 * 100 = 2500.
Addition of Exponentiations
Look at the following equation:
5x2 + 2x2 + 11x2 = 18x2
Here, each term has a similar exponent, i.e., 2, and same base x. So, when we add the constants, it gives the correct answer. Now, have a look at this:
3x2 + 5x3 + 2y2
It represents the addition of terms with different bases and exponents. To solve such problems, values of variables x and y are required.

Adding exponentiation with same bases is merely solved by counting. In, 62 + 63, we can see that the base is the same, i.e. 6. To solve this, all we can do is calculate:
62 = 6 * 6 = 36
63 = 6 * 6 * 6 = 216
Therefore,
62 + 63 = 252.
Similarly,
34 + 36 = 81 + 729 = 810.
What if the indices are same too? Well, they become easier to solve.
52 + 52 = 2(52) = 2 * 25 = 50.
Examples:
  • 32 + 53 = (3)(3) + (5)(5)(5) = 9 + 125 = 134
  • 83 + 92 = (8)(8)(8) + (9)(9) = 512 + 81 = 593
  • 32 + 32 = 2(32) = 2 * 9 = 18
  • 83 + 83 + 83 = 3(83) = 3 * 512 = 1536
Negative Exponents
It is just the opposite of positive exponents, as here, instead of repeated multiplication, you divide. Thus 3-2 = 1/32. There is a restriction that a-b = 1/ab is possible if, 'a' is non-zero.
Therefore;
2-3 = 1/23 = 1/8
5-2 = 1/52 = 1/25
Rational Exponents
An exponent, in the form of a fraction, i.e., a1/x, takes the xth root. There is no multiplication or division involved. For example, 91/2 is actually second, or square root of 2. Therefore;
41/2 = √4 = 2
81/3 = 3√8 = 2
811/4 = 4√81 = 3
Division of Exponentiation
Until now, we have seen that multiplying exponentiation is really adding exponents. So, what happens when we divide exponentiation? Do you remember, that 1/xa is actually x-a? So, now consider,
xa ÷ xb.
We can simplify it as,
xa. x-b = x(a-b)
Examples:
  • 54 ÷ 52 = 54 * 5-2 = 5(4-2) = 52 = 25
  • 123 ÷ 122 = 123 * 12-2 = 12(3 - 2) = 121 = 12
  • 166 ÷ 164 = 166 * 16-4 = 16(6-4) = 162= 256
Power of Power
Let's see another expression:
(x3)3
We can write this as,
(x3)3 = (x3)(x3)(x3)
Also,
(x3) = (x)(x)(x)
So,
(x3)3 = (x)(x)(x) (x)(x)(x) (x)(x)(x) = x9
Thus,
(xa)b = x(ab)
Examples:

  • (252)3 = 25(6) = 244140625
  • (183)5 = 18(15) = 6746640616477458432
The Zero Exponent
Anything raised to the power 0, is 1. Let us see the reason behind this. Consider,
y2 ÷ y2 = 1
Alternately,
y2 ÷ y2 = y(2 - 2) = y0
Therefore,
y0 = 1
Generalizations
an . am = a n + m
an/am = an - m
(an)m = an . am
(ab)n = an . bn
a0 = 1
a-n = 1/an
an/m = m√an
Try it Yourself!
Q.1 What is the value of 4001/2?
Q.2 Simplifly 72 * 74.
Q.3 Solve to remove the brackets: (m2)-5
Q.4 What is the value of 25k0 ?
Q.5 Write 3√g5 with single power.
Q.6 Solve 165/4 without calculator.
Q.7 Evaluate t-17 ÷ t-11
Q.8 Write h-9 as a positive fraction.
Q.9 Solve 65/62
Q.10 Simplify (f3)7