Properties of Exponents

Math has always been a mind-boggling yet interesting subject. It becomes your favorite when you get things right, and the most annoying when no matter how carefully you have solved a problem, the end answer is always different from the given solution. In mathematics, the most important part is clearing the basic concept. Especially when it comes to algebra, logarithms, etc., understanding the properties of exponents plays a vital part in problem solving.

Exponents can be defined as the digits which indicate the number of times the base value needs to be repeated. An exponent is the power written at the top right hand side of a number (base value). An example would be 2⁵, where 2 is the base value and 5 is the exponent value. The value of 2⁵ = 2 x 2 x 2 x 2 x 2 = 32. Therefore an exponent, which is also known as an index or power, is the number which represents that how many times the base value needs to be multiplied with itself.

The multiplication property states that, if the same base value with different exponents are multiplied with each other, then the exponent value can be added together. To make this statement clear, let's take this example: 5² x 5³ = (5 x 5) x (5 x 5 x 5) = 5⁵. Here, if we omit the parentheses, we will have the product of five 5s, which is written as 5⁵, which is the same as the added value of both the exponents 2 and 3. Therefore, the multiplication property of exponents is, a² x a³ = a²⁺³ = a⁵.

When you divide two different exponents having the same base, the exponents' values should be subtracted to get the correct answer. For example, 5³ / 5² = 5^3-2 = 5¹. The answer will be same if you do the long method of dividing each and every base and exponent. Therefore, remember this property and save your time. Remember, this is valid only when the base value is the same.

The power of a quotient property states that when a and b are base values wherein b is not equal to 0, and c is the exponent value provided both a and c are not 0, then, a^c/b^c = (a/b)^c. In short, if the exponent value of both the numerator and the denominator is the same, and the values mentioned above are not zero, then the exponent value for the whole fraction becomes the same.

It's true that anything multiplied by 0 is 0. But when it comes to exponential properties, anything raised to the power value 0 is '1'. Let's take an example, 5⁰ x 5¹ = 5⁽⁰⁺¹⁾ = 5¹. Now, we know that 5¹ = 5, therefore, 5⁰ x 5 = 5, meaning the value of 5⁰ being 1. PS: The value of 0⁰ is undefined.

This property comes into picture when dealing with negative exponents. It is explained with the help of this example: 5^-2 = 1/5². You can have a look at the solved example that follows, which will make this property clearer to you. 5^-2 x 5² = 5^(-2+2) = 5⁰, 5² = 25, and 5⁰ = 1. Therefore, 5^-2 = 1/25.

You can also scroll through the method of adding exponents, which again is different from the general method. We never add exponents together if there is a common base. Taking common factors aside, we need to solve the exponents and their base, step by step.

As mentioned earlier, once you have built your concept strong and clear, solving mathematics would be something that you'll look forward to. Being one of the most interesting and challenging subject, math also helps in making your brain sharp and quick.