MAT 1275: Introduction to Mathematical Analysis Dr. A. Rozenblyum II. Exponents and Radicals A. Using Integers as Exponents

Using Integers as Exponents

The notation of exponents is useful in many situations, in particular, when we work with very big numbers (for example, with distances between planets). Bellow we will show a way how exponents can be also used for very small numbers (such as, for example, as distances inside atoms). Let’s consider examples, and study some properties of exponents. Note. Notice, that in all these examples the power is equal to the number of zeros in given number.

Using Integers as Exponents Example 4. It is known that the distance from our Earth to the Sun is about 150,000,000 km (150 million kilometers). Represent this number in a short form using exponents. Note. Representation of big numbers such as in the example 4 in exponential form with the base of 10 is widely used in science. This form is called scientific notation. We will discuss scientific notation in more details latter.

Now, how about very small numbers? Consider, for example, the diameter of DNA helix. It is known that this diameter is about cm. Is it possible somehow to represent this number also in a short form using exponents? It turns out that the answer is yes. However to come up with the idea how to do this we need to develop more theory of exponents. Let’s start with some simple properties. Using Integers as Exponents Basic Properties of Exponents Working with exponents, we can operate with them like with any other numbers, in particular, make arithmetic calculation: add them, subtract, multiply, and divide. Often, we want to keep the answers also in exponential form. As for addition and subtraction, there are no special simple techniques to do that. However, for multiplication and division, in some cases we can do these operations very easily

Using Integers as Exponents Another simple but useful rule is how to raise exponents into a power.

Using Integers as Exponents The same rule is true in general case. The only restriction is that the power of numerator should be greater than the power of denominator. And, of course, all exponents must have the same base. Note. We set the restriction that a is nonzero, because if a = 0, the above formula does not make sense (we cannot divide by zero). This is the only restriction on a.

Using Integers as Exponents

Scientific Notation

Using Integers as Exponents End of the Topic