 Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum            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 Use the rule xa xb = xa-b to figure out the value of x when the exponent is 0.  If a - b = 0, what does that tell you about a and b?   Close Tip If a - b = 0, what does that tell you about a and b? Problem B2 What happens if the exponent is a negative integer like -1? Solve x3 x4 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 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 Express (x3)2 as a multiplication problem and then simplify it as much as possible. Problem B5 Consider your answers to Problems B1-B4. In each case, can 0 be a valid base? Explain why (or why not).   Session 3: Index | Notes | Solutions | Video