Rational Exponents
Prepared by: Sa’diyya Hendrickson
Name:
Date:
Until now, we’ve learned the following elementary operations: addition, subtraction, multiplication,
division and exponentiation (by integer powers). We call addition and subtraction inverse operations
because one can be used to undo the other. The same can be said about multiplication and division.
Now, consider exponentiation by a positive integer n, producing the expression bn .
Can we create an operation that undoes
or reverses exponentiation by a
power of n?
Answer: Yes! In fact, this question is the motivation behind defining rational exponents!
Below, we will conduct an investigation wherein we will attempt to uncover a practical definition for a
rational exponent, that is consistent with all of the properties of exponents previously established.
1. Consider the statement below and complete parts (a) – (c):
25 = 251
= 25(1/2 + 1/2)
= 251/2 · 251/2
by Product Property
for k = 251/2
=k·k
= k2
a) What positive number do we square (raise to the power of 2) to get 25?
b) To remain consistent with our previous definition and Properties of Exponents, we require
that k = 251/2 =
.
c) In words: 251/2 is the positive number k that when raised to the power of
answer is
.
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, the
c Sa’diyya Hendrickson
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Level: bm/n
Investigation + Definition
2. In a similar manner as Number 1 (above), read the statement below and complete parts (a) – (c).
8 = 81
= 8(1/3 + 1/3 + 1/3)
= 81/3 · 81/3 · 81/3
by Product Property
for t = 81/3
=t·t·t
= t3
a) What number do we cube (raise to the power of 3) to get 8?
b) To remain consistent with our previous definition and Properties of Exponents, we require
that t = 81/3 =
.
c) In words: 81/3 is the number t that when raised to the power of
, the answer is
.
3. Make a conjecture (guess) about the meaning of the expression b1/n
Let’s formalize our findings in the Frayer Model below.
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Level: bm/n
Generalizations
Now, we can consider the expression bm/n where m and n are integers and n > 0.
Definition: rational exponent
Consider the following cases:
C1 n is odd
C2 n is even
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Level: bm/n
Properties of nth Roots
Question: Are the Properties of Exponents previously established applicable to rational exponents?
Anwer: Yes! Recall that we created the definition of a rational exponent by assuming these properties!
Exercise: Use Properties of Exponents to deduce the following property of nth roots, which
shows that nth roots (radicals) also distribute over products.
√
√ √
n
n
ab = n a b
Exercise: TRUE or FALSE
For any b ∈ R and n ∈ N,
√
n
bn = (bn )1/n = bn/n = b1 = b
In other words, when taking the nth root of bn , is the result is always b.
Consider the counterexample
!
4
(−2)4 . Use order of operations to determine the result.
Very Important Property of nth Roots
⎧
⎪
⎨ b
√
n
bn =
⎪
⎩
|b|
if n is odd
if n is even
When there is an even exponent underneath a radical, proceed with caution when simplifying!
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©