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Learning Math Home
Number and Operation Session 3, Part B: Exponents and Logarithms
 
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Session 3, Part B:
Exponents and Logarithms (35 minutes)

In This Part: Operations with Exponents | Scientific Notation | Logarithms

As you saw earlier in this session, one way to represent each power of two is to write the base (two) raised to a power (another number). This other number is known as the exponent. An exponent tells us how many times the base is used as a factor. Exponents can simplify the calculations for such operations as multiplication and division. For example, rather than multiply 16 • 32, we can multiply 24 • 25. Let's look at how this is done.

To compute with numbers that have exponents, you need to understand how exponents work. Here are some basic rules to begin with:

 

To add or subtract numbers with exponents, the base numbers must be the same, and the exponents must also be the same:

x4 + x4 + x3 + x3 = 2x4 + 2x3

  

 

To multiply numbers with exponents, the base numbers must be the same; then we simply add the exponents. For example:

x4 • x3

x4 = x • x • x • x and x3 = x • x • x

Because multiplication is both associative and commutative, we can solve these equations as one:

x4 • x3 = x • x • x • x • x • x • x = x7

So the end result, x7, is equivalent to x4 + 3:

x4 • x3 = (x • x • x • x) • (x • x • x) = x4 + 3 = x7

This presumes that both the bases are the same. In other words, for example, we couldn't multiply 22 by 33 because the bases are not the same.

  

 

To divide numbers with exponents, the base numbers must be the same; then we simply subtract the exponents. For example:

These examples illustrate the meaning of positive-integer exponents. But what does an exponent of 0 represent?

Problem B1

Solution  

Use the rule xa xb = xa-b to figure out the value of x when the exponent is 0.


Stop!  Do the above problem before you proceed.  Use the tip text to help you solve the problem if you get stuck.
If a - b = 0, what does that tell you about a and b?   Close Tip

 

Problem B2

Solution  

What happens if the exponent is a negative integer like -1? Solve x3x4 to find out. Explain why x cannot be equal to 0.


 
 

Now let's look at what happens when the exponent is a fraction or a decimal. We know that for positive numbers greater than or equal to 1, x2 x3 x4. Is this true for exponents between 0 and 1?


 

Problem B3

Solution  

a. 

Use the rules for multiplying exponents to determine the meaning of x1/2.

b. 

How about x1/3?

c. 

Which value is greater for positive numbers greater than 1?


 

Problem B4

Solution  

Express (x3)2 as a multiplication problem and then simplify it as much as possible.


 

Problem B5

Solution  

Consider your answers to Problems B1-B4. In each case, can 0 be a valid base? Explain why (or why not).


Next > Part B (Continued): Scientific Notation

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