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In the previous set of notes, we found the following:
Re-writing the radicand 36 as 6
2
, we have the following:
2
How can we re-write a radical (such as a square root) using an exponent?
In other words, 6 to the power of 2 to the power of what will result in 6
1
or just 6. In this case, what exponent is the equivalent of a square root?
2
2
๐ฅ
1
Using the Power Rule for Exponents, when a base is taken to a power, and
then to another power, the exponents are multiplied. So 2 times what
produces 1?
So the answer is
1
2
. A square root is equivalent to an exponent of
1
2
2
2
๐
๐
= 6
So a square root is equivalent to a power of
1
2
, which is the reciprocal of
the index 2. The same is true for any radical; to express a radical as an
exponent, we simply need to take the reciprocal of the index of the radical.
Here are a few more examples of radicals and their exponent equivalents.
๐
1
3
= 5
๐
1
4
2
๐
2
1
5
= ๐ฆ
2
5
(this example uses the Power Rule for Exponents)
Converting a radical ( โ
๐
) to an exponent (๐ฅ
1
๐
)
- to write a radical as a fractional exponent, write the radicand as the
base and the reciprocal of the index as the exponent
o the radical expression โ๐ฅ
5
4
is equivalent to
5
1
4
, which
simplifies to ๐ฅ
5
4
by using the Power Rule for Exponents
o the radical expression ( โ
4
5
is equivalent to (๐ฅ
1
4
)
5
, which also
simplifies to ๐ฅ
5
4
by using the Power Rule for Exponents
- be sure that the original radical exists, otherwise the new expression
is meaningless
Converting an exponent (๐ฅ
1
๐
) to a radical (
โ
๐
- to write a fractional exponent as a radical, write the denominator of
the exponent as the index of the radical and the base as the radicand
o the expression ๐ฅ
3
5
can be written as a radical in two ways, both
of which are equivalent
3
5
= โ๐ฅ
3
5
3
5
= ( โ
5
3
o regardless of which way you choose to write the radical
expression, the index is the same (in this case 5 )
- again, it is imperative that the radical exists, otherwise the expression
is meaningless
o โโ 9 does not exist with real numbers, so it is meaningless to
write it as
1
2
o also, keep in mind the difference between
1
2
and โ 9
1
2
1
2
= โ
1
2
= โ 1 โ 9
1
2
โ
โ
e. โ 4
โ
3
2
1
2
1
8
2
3
f. โ 32
โ
2
5
64
125
โ
1
3
โ
3
2
1
2
1
2
3
8
2
3
1
4
3
2
1
2
( โ
1
3
)
2
( โ
8
3
)
2
1
( โ
4 )
3
1
2
( 1
)
2
( 2
)
2
1
( 2
)
3
1
2
1
4
1
8
1
8
1
8
1
8
๐
f.
Example 2: Simplify the following expression by converting to radical
form and/or by using Exponent Rules. Simplify completely and do NOT
leave negative exponents in your answers. If a solution does not exist in
real numbers, write DNE.
6
โ
4
3
( 3 ๐ฅ
4
3
)
6
โ
4
3
( 3 ๐ฅ
4
3
)
6
4
3
4
3
)
4
3
4
3 ( ๐ฅ
6
4
3
4
3
3
4
8
4
3
โ 8
4
โ
20
3
๐๐
๐
Change a negative
exponent to a positive
exponent by taking
the reciprocal of the
expression.
To simplify a base (๐ฅ) to a
power
( 6
) to another power
(
4
3
), we multiply the powers,
giving us ๐ฅ
8
in this case.
๐ฅ
4
3
๐ฅ
8
can be simplified using the
Quotient Rule for exponents to
get ๐ฅ
4
3
โ 8
, which is ๐ฅ
โ
20
3
Once again, to change a negative
exponent to a positive exponent
we take the reciprocal of the
expression. Since the factor
โ
20
3
is already part of a
fraction, we can take itโs
reciprocal by simply moving it
from the numerator to the
denominator.
The first thing to notice about
this expression is that ( 27 ๐ฅ
6
)
โ
4
3
is a Product to a Power, while
( 3 ๐ฅ
4
3
) is just a product.
c. โ (
๐ฅ
3
64
โ
4
3
4
โ
3
2
๐ฅ
8
) d. (
๐ฅ
12
81
3
4
๐ฅ
9
27
โ
2
3
1
4
โ
4
5
)
64
๐ฅ
3
4
3
1
4
3
2
๐ฅ
8
(๐ฅ
12
)
3
4
81
3
4
27
๐ฅ
9
2
3
1
4
1
๐ฅ
4
5
64
4
3
( ๐ฅ
3
)
4
3
1
( โ
4 )
3
โ๐ฅ
8
๐ฅ
9
( โ
81
4
)
3
(โ 27 )
2
3
(๐ฅ
9
)
2
3
1
4 ๐ฅ
4
5
( โ
64
3
)
4
๐ฅ
4
1
( 2
)
3
โ๐ฅ
8
๐ฅ
9
27
( โโ 27
3
)
2
๐ฅ
6
1
4 ๐ฅ
4
5
( 4
)
4
๐ฅ
4
1
8 ๐ฅ
8
๐ฅ
9
27
9
๐ฅ
6
1
4 ๐ฅ
4
5
256
๐ฅ
4
1
8 ๐ฅ
8
9 ๐ฅ
9
27 ๐ฅ
6
โ 4 ๐ฅ
4
5
256
๐ฅ
4
1
8 ๐ฅ
8
๐ฅ
3
3 โ 4 ๐ฅ
4
5
๐๐
๐ฅ
3 โ
4
5
12
๐๐
๐
๐๐
๐๐
๐
Once again, the expressions from Examples 2 and 3 can be difficult to
simplify completely, so be sure to spend time working on problems like
these not only in the homework, but also as you prepare for Exam #1. If
you need assistance understanding how to simplify these types of
expressions, please let me know.
Answers to Examples:
1 a. 81 ; 1b.
1
8
; 1c. ๐ท๐๐ธ ; 1 d.
9
4
; 1 e. 0 ; 1f. 1 ; 2 a.
1
27 ๐ฅ
20
3
3a. 27 ๐ฅ
22
15
; 3b. โ
1
128 ๐ฅ
13
2
; 3c. โ
32
๐ฅ
12
; 3d.
1
12
11
5
;
