6.1: The Laws of Exponents
Multiplying and Dividing Exponential Terms
Objectives
The lesson objectives for the Laws of Exponents:
- Multiplying Exponential Terms
- Dividing Exponential Terms
Introduction
If and , then the th power of ‘’ is written as:
where is called the exponent and is called the base. The term is known as a power. In other words and .
Based on this concept, there are five laws of exponents which will be illustrated in the following lesson.
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Khan Academy Simplifying Expressions with Exponents
Guidance
1. To multiply two powers with the same base, add the exponents.
2. To divide two powers with the same base, subtract the exponents.
3. To raise a power to a new power, multiply the exponents.
4. To raise a product to a power, raise each of the factors to the power.
5. To raise a quotient to a power, raise both the numerator and the denominator to the power.
The following table will summarize the above five laws of exponents:
Laws of Exponents
If and , then
- (if )
Example A
Use this law of exponents to evaluate each of the following:
a)
b)
c)
d)
Answers
a)
The answer can be taken one step further. The base is numerical so the term can be evaluated.
b)
c)
d)
Example B
Use this law of exponents to evaluate each of the following:
a)
b)
c)
d)
Answers
a)
The answer can be taken one step further. The base is numerical so the term can be evaluated.
b)
c)
d)
Example C
Use this law of exponents to evaluate each of the following:
a)
b)
c)
d)
Answers
a)
The answer can be taken one step further. The base is numerical so the term can be evaluated.
b)
c)
The answer can be taken one step further. The base is numerical so the term can be evaluated.
d)
Example D
Use this law of exponents to evaluate each of the following:
a)
b)
c)
d)
Answers
a)
b)
c)
The answer can be taken one step further. The base of each factor is numerical so each term can be evaluated. The final answer will be the product of the two answers.
d)
Example E
Use this law of exponents to evaluate each of the following.
a)
b)
c)
d)
Answers
a)
The answer can be taken one step further. The base is numerical so each term can be evaluated.
b)
The answer can be taken one step further. The denominator is numerical so the term can be evaluated.
c)
The answer can be taken one step further. The denominator and the numerator both have numerical coefficients to be evaluated.
d)
The answer can be taken one step further. The denominator and the numerator both have numerical coefficients to be evaluated.
Example F
In many of the previous examples, powers with numerical bases were evaluated by expanding the power into its factors and determining the product of the factors.
was expanded to
The product was determined:
Therefore
This concept can also be reversed. Write 32 as a power of 2.
There are 5 twos. Therefore
Use the above concept to answer the following:
a) Write 81 as a power of 3.
b) Write as a power of 3.
c) Write as a power of 2.
Answers
a)
There are 4 threes. Therefore
b)
There are 2 threes. Therefore
Apply the law of exponents for power to a power-multiply the exponents.
Therefore
c)
There are 2 twos. Therefore
Apply the law of exponents for power to a power-multiply the exponents.
Apply the law of exponents for power to a power-multiply the exponents.
Therefore
Vocabulary
- Base
- 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.
- Exponent
- 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.
- Power
- A power is simply the name given to an algebraic expression that is raised to an exponent. and are both examples of a power.
Guided Practice
1. Perform the following operations:
i)
ii)
iii)
2. Perform the following operations:
i)
ii)
iii)
3. Perform the following operations:
i)
ii)
iii)
Answers
1. Keep the base of ‘’ and add the exponents.
i)
Multiply the coefficients , keep the base of ‘’ and add the exponents.
ii)
iii)
2. In the numerator, keep the base of ‘’ and add the exponents.
i)
Keep the base of ‘’ and subtract the exponents.
or
or
ii) Keep the base of ‘’ and subtract the exponents.
iii) Keep the base of ‘’ and subtract the exponents.
3) Keep the base of ‘’ and multiply the exponents of each factor by 3.
i) Evaluate .
ii) Keep the base of ‘’ and multiply the exponents of each factor by 7.
iii) Keep the base of ‘’ and multiply the exponents of each factor in the numerator by 3 and the exponents of each factor in the denominator by 3.
Evaluate and
Summary
In this lesson you have learned to apply five laws of exponents. These laws were:
In addition to learning these laws of exponents you also learned to write numbers as a power of another number. For example, you learned to write 8 as a power of 2. . This concept is very important in solving exponential equations. You will learn this is a future lesson in this chapter.
Problem Set
Express each of the following as a power of 3. Do not evaluate.
Apply the laws of exponents to evaluate each of the following without using a calculator. (Show Your Work)
Use the laws of exponents to simplify each of the following.
Answers
Express each of the following...
Apply the laws of exponents to evaluate...
Use the laws of exponents to...