Exponents are generally known as powers where the base number is multiplied by the power of itself to get the result. For example, 55 means that we have to multiply the base number 5 to itself 5 times. Thus, 56 = 5 x 5 x 5 x 5 x 5 x 5 this is equal to 15625. Here the number 5 is the base to which power is assigned. So the power or the superscript on the base is termed as an exponent. The expression above can be denoted as five to the power of six.
Sometimes, the power of two can be called upon as squared and the power of three can be mentioned as cubed. They are often used to find the area or volume of certain shapes. Now, there are times when we use unknown variables like x and y within the equation. Like x3 + y4 suggests that x is multiplied three times by itself and y is multiplied four times by itself. Thus after getting the value of the variables, we add them up to find the possible result. In this article, we would like to find out the properties of exponents which are listed below.
Laws of Exponents
There are around seven properties or rules of exponents. Each rule is dedicated to finding the solution for different equations and how to add, subtract, multiply and divide using exponents. Here we would take two variables x and y with powers and b to suffice with examples.
Property of Product rule
According to the formula, xa * xb = xa+b. Let us take a simple example to prove this. We take a base number as 3 and the powers or exponents as 2 and 3 for a and b respectively. Thus the equation looks like this:
52 + 53 = 52+3 =55 = 5 x 5 x 5 x 5 x 5 = 3125. This is generally called the law of product. Here we are adding the exponents for the same base number.
Property of Quotient rule
The formula for the quotient rule is denoted by the equation, xa/xb = xa-b. Let us find the value using a simple example by taking the base number as 3 and the powers a and b as 5 and 2 respectively. The equation then looks like this:
35/32 = 35-2 = 33 = 3 x 3 x 3 = 27. This is known as the law of quotient. Here we subtract the exponents for the same base number.
Property of Zero rule
The property can be formulated by the equation, x0 = 1. With a simple example, we can prove this by taking a similar base number with similar powers and we divide them. The equation would look like this:
23/23 = 23-3 = 20 = 1. This is called the law of zero where we get the value as 1 no matter how big the equation is.
Property of Negative rule
The formula for this property can be equated as x-3=1/x3. Here the exponent is flipped over as a reciprocal to get a positive exponent number. For example, 2-3 = ½3 = 1/6.
Property of power of a power rule
According to the formula, it states that (xa)b = xab. We can easily use this with an example. We use a base number as 2 and the exponents, a and b as 3 and 4 respectively.
So, (23)4 = (2 x 2 x 2) (2 x 2 x 2) (2 x 2 x 2) (2 x 2 x 2) = 8 x 8 x 8 x 8 = 4096. Similarly, if we use the formula, then (23)4=(23×4) = 212 = 4096
Property of power of a product rule
The formula states that, (x * y)a = xaya. It is like multiplying the bases with the same power.
Property of power of a quotient rule
According to the formula, (x/y)a = xa/ya. It is like having different bases with the same power.
These are the few exponent rules presented above. For more information, do visit Cuemath, an online platform that excels in teaching math and coding.
