Exponent Rules
Through the rules of exponents, you can calculate values with high powers. Many of the arithmetic operations like addition, subtraction, multiplication, and division can be conveniently be performed in quick steps using the exponential rules. For using these exponential rules across two exponents, the base of the exponents should be equal. Further, the rules also help in simplifying single exponents with complex powers involving fractions, decimals, roots.
Here the laws of exponents are helpful to easily understand exponents with exponent values of 0, and 1. Through this webpage, we can understand the different rules of exponents, involving different kinds of numbers for the base and exponents.
1.  What are Exponent Rules? 
2.  Different Exponent Rules 
3.  Power of Exponent Rules 
What are Exponent Rules?
We have a set of rules of exponents (or) laws of exponents which make the process of simplifying exponents easy. The rules of exponents are used to simplify exponents, especially in algebra. Solving two or more exponents is possible through the use of the exponent rules. These rules are helpful to simplify the exponents having decimals, fractions, irrational numbers, and negative integers.
Some of the basic rules of exponents involving addition between two exponents is a^{m} × a^{n} = a^{m + n}. Here in this below content, we shall look into similar exponent rules with the division between two terms, which results in the subtraction of exponents. Also, the values of exponents with 0 and 1 can be conveniently calculated with the help of these exponent rules.
Different Exponent Rules
For the sake of convenience, the exponent rules have been named differently, to help for the various arithmetic operations The various exponent rules are the product property, quotient property, zero property of exponents, and negative property of exponents. These exponent rules covered below are listed as follows.
 a^{m} × a^{n} = a^{m + n}
 a^{m}/a^{n} = a^{m  n}
 a^{0} = 1
 a^{m} = 1/a^{m}
Let us look into the details of each of the exponent rules.
Product Property of Exponents
The product property of exponents is used to multiply expressions with the same bases. This property says, "To multiply two expressions with the same base, add the exponents while keeping the base the same." This rule involves adding exponents with the same base. Here the rule is useful to simplify two exponents with a multiplication operation between them.
Here is an example.
Using the law  Without using the law 
2^{3} × 2^{5} = 2^{(3 + 5)} = 2^{8}  2^{3} × 2^{5} = (2 × 2 × 2) × (2 × 2 × 2 × 2 × 2) = 2^{8} 
We can conclude that without using the law, the solving of expression take more steps and more calculation work.
Quotient Property of Exponents
The quotient property of exponents is used to divide expressions with the same bases. This property says, "To divide two expressions with the same base, subtract the exponents while keeping the base same." This is helpful to solving an exponent, without actually performing the division process. The only required condition is that the two exponents should have the same base.
Here is an example.
Using the law  Without using the law 
2^{5}/2^{3} = 2^{5  3} = 2^{2}  2^{5}/2^{3} = (2 × 2 × 2 × 2 × 2)/(2 × 2 × 2 ) = 2^{2} 
We can clearly see that without using the law, the solving of expression take more steps and more calculation work.
Zero Property of Exponents
The zero property of exponents is applied when the exponent of any base is 0. This property says, "Any number (other than 0) raised to 0 is 1." Note that 0^{0} is not defined. The will help us understand that irrespective of the base the value for a zero exponent is always equal to 1.
Here is an example.
Using the law  Without using the law 
2^{0} = 2  2^{0} = 2^{5  5}^{ }= 2^{5}/2^{5} = (2 × 2 × 2 × 2 × 2)/(2 × 2 × 2 × 2 × 2) = 1 
Using the law we simply have 2^{0} = 2. Alternatively, without using the law we can understand the same law with more number of steps: 2^{0} = 2^{5  5}^{ }= 2^{5}/2^{5} = (2 × 2 × 2 × 2 × 2)/(2 × 2 × 2 × 2 × 2) = 1
Negative Property of Exponents
The negative property of exponents is used when an exponent is a negative number. This property says, "To convert any negative exponent into positive exponent, the reciprocal should be taken." The expression is transferred from the numerator to the denominator with the change in sign of the exponent values.
Here is an example.
Using the law  Without using the law 
2^{2} = 1/2  2^{2} = 2^{02} = 2^{0}/2^{2} = 1/2^{2} 
Using the law, we can solve it in just one go, like 2^{2} = 1/2. Alternatively without using the law we have 2^{2} = 2^{02} = 2^{0}/2^{2} = 1/2^{2}
Power of Exponent Rules
The power of exponent rules can be understood with the ease of doing the calculation. Basic exponent rules can be further expanded to include different bases having the same exponent values. The rules involving different bases for the exponents are as follows.
 (a^{m})^{n} = a^{mn }
 (ab)^{m} = a^{m} b^{m}
 (a/b)^{m} = a^{m}/b^{m}
Let us understand the power of exponent for each of the formulas.
Power of a Power Property of Exponents
The power of a power property of exponents is used to simplify expressions of the form (a^{m})^{n}. This property says, "When we have a single base with two exponents, just multiply the exponents." The two exponents are available one over the other. These can be conveniently multiplied to make a single exponent.
Here is an example.
Using the law  Without using the law 
(2^{2})^{3} = 2^{6}  (2^{2})^{3} =(2^{2}).(2^{2}).(2^{2}) =(2.2).(2.2).(2.2) = 2^{6} 
Using the law, (2^{2})^{3} = 2^{6}, whereas, the same thing without the exponent rules take more steps and is expressed as (2^{2})^{3} =(2^{2}).(2^{2}).(2^{2}) =(2.2).(2.2).(2.2) = 2^{6}
Power of a Product Property of Exponents
The power of a product property of exponents is used to find the result of a product that is raised to an exponent. This property says, "Distribute the exponent to each multiplicand of the product." Here the base of the expression is the same and the power is different.
Here is an example.
Using the law  Without using the law 
(xy)^{3} = x^{3}.y^{3}  (xy)^{3} =(xy).(xy).(xy) = (x.x.x).(y.y.y) x^{3}.y^{3} 
Using the law, (xy)^{3} = x^{3}.y^{3}. On the other hand, the same thing can be expressed in multiple steps, without using the law. (xy)^{3} =(xy).(xy).(xy) = (x.x.x).(y.y.y) x^{3}.y^{3}
Power of a Quotient Property of Exponents
The power of a quotient property of exponents is used to find the result of a quotient that is raised to an exponent. This property says, "Distribute the exponent to both the numerator and the denominator." Here the bases are different with the division between them and the exponents are the same for both the bases.
Here is an example of the above exponent rule.
Using the law  Without using the law 
(x/y)^{3} = x^{3}/y^{3}  (x/y)^{3} = x/y . x/y . x/y = x^{3}/y^{3} 
We can use the law and simply solve it as, (x/y)^{3} = x^{3}/y^{3}, and we can also solve the same thing without the law and it involved multiple steps. (x/y)^{3} = x/y . x/y . x/y = x^{3}/y^{3}.
Now that you have learned all these properties, you have an idea of how and why a particular law works and how these laws of exponents make our work easier. Let's try and solve a few examples using these laws.
Important Notes
The laws of exponents are as follows:
 a^{0 }= 1
 a^{1 }= a
 a^{m }× a^{n }= a^{m+n}
 a^{m}/a^{n }= a^{mn}
 a^{m }= 1/a^{m}
 (a^{m})^{n }= a^{mn}
 (ab)^{m }= a^{m}b^{m}
 (a/b)^{m }= a^{m}/b^{m}
Tips and Tricks
 If a fraction has a negative exponent, then we take the reciprocal of the fraction to make the exponent positive, i.e. (a/b)^{m} = (b/a)^{m}
 When the exponents are the same, we can set the bases equal and vice versa, i.e. a^{m} = a^{n}, m =n
 We can convert a radical into an exponent using the following property: a^{1/n} = \(\sqrt[n]{a}\)
Exponent Rules Solved Examples

Example 1: The weight of a caterpillar when its larva hatches is about 10^{3} grams. It can eat 10^{4} times its body weight every day. How many grams of food can the larva eat on the day it hatches?
Solution:
The weight of the caterpillar on the day it hatches = 10^{3} grams. Since it eats 10^{4} times its body weight, the amount it eats on the day it hatches is: 10^{3} × 10^{4}=10^{(3 + 4)} = 10^{1} = 10. Therefore, the amount of food the larvae eats on the day it hatches is 10 grams.

Example 2: If an asteroid is 10^{15} miles away from the earth and it is traveling at a speed of 10^{7} miles per day, how long (in days) will it take the asteroid to reach earth? Indicate your answer in the form of an exponent.
Solution:
Given that the distance between the earth and the asteroid = 10^{15} miles. The speed of the asteroid = 10^{7} miles/day. We know that: Time= Distance/Speed = 10^{15}/10^{7}= 10^{15  7} = 10^{8}. Therefore, the time taken by the asteroid to reach the earth is 10^{8} days.
Practice Questions on Exponent Rules
FAQs on Exponent Rules
What is the Rule For Zero Exponents?
The rule of zero exponents is a^{0}=1. Here, 'a' the base can be any number other than 0. This law is helpful in simplifying this without any confusion.
Can the Base of an Exponential Function be Negative?
Yes, the base of an exponential function can be negative. Some of the quick examples with bases with negative values are Example: If f(x)=(2), then find f(3). Here we have f(3)=(2)^{3} = 2 × 2 × 2 =8.
What are the Four Rules of Exponents?
The four rules of exponents involve multiplication between exponents, the division between exponents, base with an exponent of 1, and base with an exponent of 0. These four laws are helpful to derive other rules of exponents. The four rules of exponents are as follows. a^{m} × a^{n} = a^{m + n}; a^{m}/a^{n} = a^{m  n}; a^{0} = 1; a^{m} = 1/a^{m}
What is the Difference Between Exponents and Powers?
The exponents are powers refer to the same term. A number of the form a^{m} has a base 'a' and a power m. m is also referred to as an exponent. The rules involving exponents of 0 and 1 are a^{0} = 1, and a^{1} = a.
What is the Purpose of the Exponent Rules?
The purpose of exponent rules is to simplify the expressions, and also to write the expressions in fewer steps. Let us understand more of this with a simple example. Without the exponent rules the expression 2^{3} × 2^{5} is written as (2 × 2 × 2) × (2 × 2 × 2 × 2 × 2) = 2^{8}. Further is the same expression with the help of exponent rules can be simplified in two quick steps as 2^{3} × 2^{5} = 2^{(3 + 5)} = 2^{8}.
How to Prove Exponent Rules?
The exponent rules can be proved easily by expanding the exponential terms. The exponential expression is expanded by writing the base as many times as the power value. The exponent of the form a^{n} is written as a × a × a × a × a × .... n times. Further on multiplying we can obtain the final value of the exponent.
Can the Exponent be a Faction?
The exponent value can also be a fraction. The rules relating to the fraction exponent value is (a^{m})^{1/n} = a^{m/n}. This rule is sometimes helpful to simplify and transform a surd into an exponent.
How are Exponent Rules Used in Algebra?
The exponent rules are very helpful in each of the algebra formulae. For example, the algebraic formula of (a + b)^{2} = a^{2} + b^{2} + 2ab can be written and calculated easily by applying the rules of exponents.