Suppose you have the expression:
How could you write this expression in a more concise way?
James Sousa: Simplify Exponential Expressions- Quotient Rule
In the expression x3, the x is called the base and the 3 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.
m factors↑aman=(a×a×…×a)←→−−−−−−−−−−(a×a×…×a)←→−−−−−−−−−− m>n;a≠0↓ n factorsaman=(a×a×…×a)←→−−−−−−−−−−↓ m−n factorsaman=am−n
RULE: To raise a quotient to a power, raise both the numerator and the denominator to the power.
(ab)n=ab×ab×…×ab←→−−−−−−−−−− ↓n factorsn factors ↑(ab)n=(a×a×…×a)←→−−−−−−−−−−(b×b×…×b)←→−−−−−−−−− ↓n factors(ab)n=anbn (b≠0)
27÷2327−324The base is 2.Keep the base of 2 and subtract the exponents.The answer is in exponential form.
The answer can be taken one step further. The base is numerical so the term can be evaluated.
x8x2x8−2x6x8x2=x6The base is x.Keep the base of x and subtract the exponents.The answer is in exponential form.
16x5y54x2y34(x5y5x2y3)4x5−2y5−34x3y2The bases are x and y.Divide the coefficients - 16÷4=4. Keep the base of x and y andsubtract the exponents of the same base.
Concept Problem Revisited
x⋅x⋅x⋅x⋅x⋅x⋅x⋅x⋅x⋅y⋅y⋅y⋅y⋅yx⋅x⋅x⋅x⋅x⋅x⋅y⋅y⋅y can be rewritten as x9y5x6y3 and then simplified to x3y2.
- In an algebraic expression, the base is the variable, number, product or quotient, to which the exponent refers. Some examples are: In the expression 25, ‘2’ is the base. In the expression (−3y)4, ‘−3y’ 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 25, ‘5’ is the exponent. It means to multiply 2 times itself 5 times as shown here: 25=2×2×2×2×2.
- In the expression (−3y)4, ‘4’ is the exponent. It means to multiply −3y times itself 4 times as shown here: (−3y)4=−3y×−3y×−3y×−3y.
- 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: 2325
- Evaluate the numerator and denominator separately and then simplify the fraction: 2325
- Use your result from the previous problem to determine the value of a: 2325=12a
- Use your results from the previous three problems to help you evaluate 2−4.
Answers for Explore More Problems
To view the Explore More answers, open this PDF file and look for section 6.2.