Exponent Notation: Definition and Rules
In this lesson, you will learn about the exponent notation, its components and how to calculate it, as well as extract the most important properties of powers used to simplify algebraic expressions. Power, quite simply, is multiplying a number of times by itself. For example, we take the power as the product result of , where the small, raised number to the right of the base number indicates the number of times the base number is multiplied by itself.
Example1:
Let's calculate the following powers:
Now lets do the inverse, transform the follwing numbers to exponnent notation:
Exponential Notation Rules
The Product Rule
From the above definition and examples, we can easily calculate the product of the two powers. we've got:
How do you think we got the power that the exponent is ?
You will no doubt have noticed that if we add the exponents of the powers and , we get 6. From this example, we can deduce the following property:
Example2:
Multiply each expression using the product rule:
Solution:
The Quotient Rule
From the above definition and examples, we can easily calculate the quotient of the two powers. we've got:
How do you think we got the power that the exponent is ?
In this example, we have subtracted the exponent of the power in the denominator from the exponent of the power in the numerator (). From this example, we can deduce the following property:
Example3:
Divide each expression using the quotient rule:
Solution:
The Products-to-Powers Rule for Exponents
In this section, the rule of exponents applies when we are raising a product to a power. Let's take the following example:
The base of the power is the product .
applying the formula in the definition:
Now, how did we get the answer ?
If you notice, we've removed the parentheses and raised both factors, and , to the fourth power. From this example, we can deduce the following property:
Example4:
- Simplify:
- Simplify:
Solution:
The Quotients-to-Powers Rule for Exponents
In this section, the rule of exponents applies when we are raising a quotient to a power. Let's take the following example:
The base of the power is the quotient .
applying the formula in the definition:
Now, how did we get the answer ?
If you notice, we've removed the parentheses and raised denominator and numerator, and , to the fourth power. From this example, we can deduce the following property:
Example5:
- Simplify:
- Simplify:
Solution:
The Negative-Exponent Rule
In the definition of the exponontial notation, We saw that the exponent is a positive integer, but what about if you see a negative exponent? We can easily calculate a power with negative exponent follwing the below example: We simplify the quotient \frac{5^3}{5^5} with two different ways.
This means that . A negative exponent in the numerator becomes a positive exponent when moved to the denominator and vice versa.
Example6:
Write each expression without negative exponents, and evaluate if possible
Solution:
The Zero-Exponent Rule
There is a special case in calculating powers, when the exponent is zero. How can we calculate a power taht zero as an exponent? As in the previous paragraphs, we start from the following two examples:
This means that
Example7:
Here are examples involving simplification using the zero-exponent rule:
Things to remember about exponents
There are some special cases which are very important in simplifying algebraic expressions. We can prove these special cases in the same way in the previous proofs.