Suppose you have the expression:
How could you write this expression in a more concise way?
In the expression , the is called the base and the is called the exponent . Exponents are often referred to as powers . When an exponent is a positive whole number, it tells you how many times to multiply the base by itself. For example:
There are many rules that have to do with exponents (often called the Laws of Exponents ) that are helpful to know so that you can work with expressions and equations that involve exponents more easily. Here you will learn two rules that have to do with exponents and quotients.
RULE: To divide two powers with the same base, subtract the exponents.
RULE: To raise a quotient to a power, raise both the numerator and the denominator to the power.
The answer can be taken one step further. The base is numerical so the term can be evaluated.
Concept Problem Revisited
can be rewritten as and then simplified to .
- In an algebraic expression, the base is the variable, number, product or quotient, to which the exponent refers. Some examples are: In the expression , ‘2’ is the base. In the expression , ‘ ’ is the base.
- In an algebraic expression, the exponent is the number to the upper right of the base that tells how many times to multiply the base times itself. Some examples are:
- In the expression , ‘5’ is the exponent. It means to multiply 2 times itself 5 times as shown here: .
- In the expression , ‘4’ is the exponent. It means to multiply times itself 4 times as shown here: .
- Laws of Exponents
- The laws of exponents are the algebra rules and formulas that tell us the operation to perform on the exponents when dealing with exponential expressions.
Simplify each of the following expressions.
Simplify each of the following expressions, if possible.
- Simplify using the laws of exponents:
- Evaluate the numerator and denominator separately and then simplify the fraction:
- Use your result from the previous problem to determine the value of :
- Use your results from the previous three problems to help you evaluate .