Teaching Notes 8.15: Using Irrational Numbers as Exponents
MultiplicationandDivision........................... 6.3: Using the Properties of Exponents That Apply to
exponents. A common mistake students make with these problems is to express one number in
terms of the base of another number. Other common mistakes include applying the properties of
exponents incorrectly.- Explain that rational numbers can be expressed as fractions; irrational numbers cannot be
expressed as fractions. Some examples of rational numbers include 3, 0.24,−21, and 7.253.
Examples of irrational numbers include
√
2,
√
5, andπ.- Provide this example of a base with a rational exponent: 2^3 = 2 · 2 · 2 =8. Now provide
this example of a base with an irrational exponent: 2πequals 2 multiplied by itselfπtimes.
Becauseπis an irrational number (it is a nonterminating, nonrepeating decimal), 2πhas no
exact value. - Review the information and examples on the worksheet with your students. Explain that
the properties of exponents apply to both rational and irrational exponents. You may find it
helpful to review 6.3: ‘‘Using the Properties of Exponents That Apply to Multiplication and
Division.’’ Emphasize that the base of the numbers must be the same before students can
apply the properties of exponents. If the bases are different, students must rewrite one or
both of the numbers so that the bases are the same. Be sure to discuss the examples fully. In
particular, note that in the second example, 49 was rewritten as 7^2 , and in the third example,
both 8 and 16 must be rewritten as a power of 2. In these examples, the problems had to be
rewritten so that the bases are the same.
EXTRA HELP:
Be sure the bases are the same before you apply the properties of exponents.ANSWER KEY:
(1) 42√ 2
(2) 32 = 9 (3) 5√ 2 − 2
(4) 26 +^2√ 2
or 4^3 +√ 2
(5) 62√ 7 − 2
(6) 54 −^3√ 2
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(Challenge)Leah is incorrect. She tried to multiply the expressions by adding the exponents
but the bases are different. She should have rewritten 9√ 2
as 3^2√ 2. The correct answer is 3^3
√ 2
.
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