Teaching Notes 8.14: Using Rational Numbers as Exponents
Rational numbers have a special meaning when they are used as exponents. The denominator
indicates the root of the number and the numerator indicates the number of times the root is
used as a factor. Students often confuse the meaning of the denominator with the meaning of the
numerator. They may also fail to realize that negative exponents represent reciprocals.- Review the meaning of the roots of numbers by providing the following examples. The square
root of 36 is 6, which can be written as
√
36 =6. Note that finding the square root of a num-
ber is the opposite of squaring the number, 6^2 =36. Also review that the cube root of 8 is 2,
which can be written as^3√
8 =2. Finding the cube root of a number is the opposite of cubing
the number, 2^3 =8.- Explain to your students that the roots of numbers can be expressed with exponents. For
example, 36
1(^2) can be expressed as
√
- Another example is 8
1(^3) =^3
√
- The denominator of
the exponent indicates the root of the number. If there is no number in the indentation of
the radical symbol, the square root is implied. Emphasize that the numerator indicates the
number of times the root is used as a factor. - Review the information and examples on the worksheet with your students. Explain that
exponents may be either positive or negative. If the exponent is negative, students must
write the reciprocal of the expression and use the positive exponent. The meanings of the
numerator and denominator do not change.
EXTRA HELP:
Remember to write a reciprocal if the exponent is negative.ANSWER KEY:
(1) 343 (2) 2 (3)1
25
(4)1,000 (5)
1
8
(6)
1
100
(7) 64 (8)
1
36
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(Challenge)Both methods are correct. In this case, it is easier to simplify the exponent first.
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