# 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

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?

**b**, where b is called the

^{n}**base**and 'n' is known as the

**exponent or index or power**.

Well, exponentiation is a mathematical operation involving numbers in the form a

^{b}, 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:

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".

- a
^{2}as a squared (derived from area of a square is l.l or l squared) - b
^{3}as b cubed (derived from volume of cube is l.l.l or l cubed)

^{2}is "four squared" and 5^{3}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:

are all positive exponents. It is important to note here that x

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:

Now, when we say addition of exponents, it actually is multiplication of two exponentiations.

Now, for multiplying them together;
Also,

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, 3

There exists an exception. In case the bases are different and the powers are same, you

**2**

^{6}, 34^{5}and 678^{9}^{4}and 5^{3}, when expanded, will become (x)(x)(x)(x) and (5)(5)(5). Moreover, 6x^{2}will be**6(x)(x) and not (6x)(6x)**

**Examples:****a**^{3}= (a)(a)(a)**3a**^{2}= 3(a)(a)**(3a)**^{2}= (3a)(3a)

**a**^{2}b = (a)(a)(b)**2ab**^{2}= 2a(b)(b)**2a**^{2}b^{2}= 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:

**x**^{y}is exponentiation where y is the exponent.Now, when we say addition of exponents, it actually is multiplication of two exponentiations.

**x**

^{(2 + 3)}= x^{2}. x^{3}= x^{5}**x**

^{2}= (x)(x)**x**

^{3}= (x)(x)(x)**x**

^{2}. x^{3}= (x)(x)(x)(x)(x)**6**

^{3}. 6^{4}= 6^{(3 + 4)}= 6^{7}= 279936**Examples:****5**^{2}. 5^{4}= 5^{2 + 4}= 5^{6}= 15625**Check:****5**^{2}= (5)(5) = 25**5**^{4}= (5)(5)(5)(5) = 625**25 * 625 = 15625**

**7**^{3}. 7^{5}= 7^{3 + 5}= 7^{8}= 5764801**Check:****7**^{3}= (7)(7)(7) = 343**7**^{5}= (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, 3

^{4}+ 5^{2}= ?. 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 3^{4}= 81 and 5^{2}= 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.**(a**

^{4})(b^{4})**= (a)(a)(a)(a)(b)(b)(b)(b)**

**= (ab)(ab)(ab)(ab)**

**= (ab)**

^{4}**Examples:****(4**^{3})(5^{3}) = (4 * 5)^{3}= 20^{3}= 8000**Check****(4**^{3}) = 64**(5**^{3}) = 125**64 * 125 = 8000**

**(10**^{2})(5^{2}) = (10 * 5)^{2}= 50^{2}= 2500**Check****(10**^{2}) = 100**(5**^{2}) = 25**25 * 100 = 2500.**

Addition of Exponentiations

Look at the following equation:

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:
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, 6

Therefore,
Similarly,
What if the indices are same too? Well, they become easier to solve.

**5x**

^{2}+ 2x^{2}+ 11x^{2}= 18x^{2}**3x**

^{2}+ 5x^{3}+ 2y^{2}Adding exponentiation with same bases is merely solved by counting. In, 6

^{2}+ 6^{3}, we can see that the base is the same, i.e. 6. To solve this, all we can do is calculate:**6**

^{2}= 6 * 6 = 36**6**

^{3}= 6 * 6 * 6 = 216**6**

^{2}+ 6^{3}= 252.**3**

^{4}+ 3^{6}= 81 + 729 = 810.**5**

^{2}+ 5^{2}= 2(5^{2}) = 2 * 25 = 50.**Examples:****3**^{2}+ 5^{3}= (3)(3) + (5)(5)(5) = 9 + 125 = 134**8**^{3}+ 9^{2}= (8)(8)(8) + (9)(9) = 512 + 81 = 593**3**^{2}+ 3^{2}= 2(3^{2}) = 2 * 9 = 18**8**^{3}+ 8^{3}+ 8^{3}= 3(8^{3}) = 3 * 512 = 1536

Negative Exponents

It is just the opposite of positive exponents, as here, instead of repeated multiplication, you divide. Thus 3

Therefore;

^{-2}= 1/3^{2}. There is a restriction that a^{-b}= 1/a^{b}is possible if, 'a' is non-zero.Therefore;

**2**

^{-3}= 1/2^{3}= 1/8**5**

^{-2}= 1/5^{2}= 1/25Rational Exponents

An exponent, in the form of a fraction, i.e., a

^{1/x}, takes the xth root. There is no multiplication or division involved. For example, 9^{1/2}is actually second, or square root of 2. Therefore;**4**

^{1/2}= √4 = 2**8**

^{1/3}=^{3}√8 = 2**81**

^{1/4}=^{4}√81 = 3Division 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/x
We can simplify it as,

^{a}is actually x^{-a}? So, now consider,**x**

^{a}÷ x^{b}.**x**

^{a}. x^{-b}= x(^{a-b})**Examples:****5**^{4}÷ 5^{2}= 5^{4}* 5^{-2}= 5(^{4-2}) = 5^{2}= 25**12**^{3}÷ 12^{2}= 12^{3}* 12^{-2}= 12(^{3 - 2}) = 12^{1}= 12**16**^{6}÷ 16^{4}= 16^{6}* 16^{-4}= 16(^{6-4}) = 16^{2}= 256

Power of Power

Let's see another expression:
We can write this as,
Also,
So,
Thus,

**(x**

^{3})^{3}**(x**

^{3})^{3}= (x^{3})(x^{3})(x^{3})**(x**

^{3}) = (x)(x)(x)**(x**

^{3})^{3}= (x)(x)(x) (x)(x)(x) (x)(x)(x) = x^{9}**(x**

^{a})^{b}= x(^{ab})**Examples:****(25**^{2})^{3}= 25(^{6}) = 244140625**(18**^{3})^{5}= 18(^{15}) = 6746640616477458432

The Zero Exponent

Anything raised to the power 0, is 1. Let us see the reason behind this. Consider,
Alternately,
Therefore,

**y**

^{2}÷ y^{2}= 1**y**

^{2}÷ y^{2}= y(^{2 - 2}) = y^{0}**y**

^{0}= 1Generalizations

**a**

^{n}. a^{m}= a^{n + m}**a**

^{n}/a^{m}= a^{n - m}**(a**

^{n})^{m}= a^{n}. a^{m}**(ab)**

^{n}= a^{n}. b^{n}**a**

^{0}= 1**a**

^{-n}= 1/a^{n}**a**

^{n/m}=^{m}√a^{n}Try it Yourself!

**Q.1**What is the value of 400

^{1/2}?

**Q.2**Simplifly 7

^{2}* 7

^{4}.

**Q.3**Solve to remove the brackets: (m

^{2})

^{-5}

**Q.4**What is the value of 25k

^{0}?

**Q.5**Write

^{3}√g

^{5}with single power.

**Q.6**Solve 16

^{5/4}without calculator.

**Q.7**Evaluate t

^{-17}÷ t

^{-11}

**Q.8**Write h

^{-9}as a positive fraction.

**Q.9**Solve 6

^{5}/6

^{2}

**Q.10**Simplify (f

^{3})

^{7}