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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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