RATIONAL EXPONENTS
Algebra One
Explore in your Notebook…
• Evaluate the following:
a) 121 b) 15
c) – 81
e) 6.25
f) 0.04 g)
d) 2
9
h)
25
784
• What is an irrational number?
• What is a rational number?
• How can we predict/determine when a
square root is irrational or rational?
• List the first 15 perfect squares.
Perfect Squares
It’s important to know your perfect squares –
they can be useful for estimating values for
irrational square roots.
Estimate the following square roots without a
calculator:
14
35
110
IMPORTANT FACT
SQUARING and SQUAREROOTING are INVERSE
OPERATIONS.
Can we take roots other than “square
roots”?
Yes we can take any kind of root – for example:
The cube root of 8 is noted as
3
The fifth root of 32 is noted as 5
8
32
SQUARE ROOTS & Nth ROOTS
Yes we can take any kind of root – for example:
The square root of any number:
The nth root of any number:
n
2
a =a
n
a =a
*the “index” of the radical tells you what root
you are taking, if you don’t see an “n” then it
is square root.
What does this have to do with
EXPONENTS?
Consider… roots and powers are inverse
operations
a =a
↔
Cube root ↔ cubing
a =a
Fourth root ↔ fourth power a = a
Nth root ↔ nth power
a =a
square root
2
squaring
3
3
4
n
n
4
If you could turn a “root” into a
power, what would it look like?
Remember – inverse operations “cancel” out.
2
a =a
3
4
n
so
3
so
4
so
a =a
a =a
n
a =a
so
3
4
n
a
2
a
3
a
4
a
n
is the same as
is the same as
is the same as
is the same as
RATIONAL EXPONENTS
The nth root of a positive number can be
written as a power with base “a” and
exponent “1/n”
n
a =a
1/ n
RATIONAL EXPONENTS
4
1/ 4
81 = 81
3
1/ 3
216 = 216
1/ 2
289 = 289
This makes nth roots very easy to evaluate on our
calculator, just remember to put parentheses
around the full exponent.
Get comfortable going back & forth between
radical & exponential notation for nth roots.
• Write the following using rational exponent
notation:
a)
b)
• Write the following using radical notation.
c)
d) 61/9 e) Seventh root(s) of 13
What about Rational Exponents that do NOT
have a numerator of ONE?
• What does this mean? 163/4
consider reversing the power of a power
property 163/4 = 1631/4 = (163)1/4
So what does that numerator represent?
RATIONAL EXPONENTS
The nth of a positive number can be written as
a power with base “a” and exponent “1/n”
n
m
a =
(a)
n
m
=a
m/n
This makes it quite easy to evaluate
on your calculator if you remember
how to rewrite them!
(
4
16 ) =
3
( 9)
5
=
(
27 ) =
(
256 ) =
3
4
4
3
Cool Down – THINK ABOUT IT…
What is the meaning of a negative rational
exponent?
-4/3
8
Homework
Worksheet – Radicals & Rational Exponents