Radical Expressions And Rational Exponents Worksheet

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Radical Expressions and
8-6
Rational Exponents
TEKS 2A.2.A Foundations for functions: use tools including factoring and
properties of exponents to simplify expressions … .
Who uses this?
Objectives
Rewrite radical
Guitar makers use radical expressions to
expressions by using
ensure that the strings produce the correct
rational exponents.
notes. (See Example 6.)
Simplify and evaluate
radical expressions and
You are probably familiar with finding the
expressions containing
square and square root of a number. These two
rational exponents.
operations are inverses of each other. Similarly,
Traditional mariachi bands are
there are roots that correspond to larger powers.
popular throughout Texas.
Vocabulary
index
(
)
-5
= 25 and
-5
2
= 25.
2
5
and
are
square
roots of 25 because
5
rational exponent
= 8.
3
2
is the
cube
root of 8 because
2
(
)
4
-2
= 16 and
-2
= 16.
4
2
and
are
fourth
roots of 16 because
2
= b.
n
a
is the
nth
root of b if
a
n
The nth root of a real number a can be written as the radical expression
a ,
where n is the index (plural: indices) of the radical and a is the radicand.
When a number has more than one real root, the radical sign indicates only the
When a radical sign
shows no index, it
principal, or positive, root.
represents a square
root.
Numbers and Types of Real Roots
Case
Roots
Example
Odd index
1 real root
The real 3rd root of 8 is 2.
The real 4th roots of 16 are ±2.
Even index; positive radicand
2 real roots
-16 has no real 4th roots.
Even index; negative radicand 0 real roots
Radicand of 0
1 root of 0
The 3rd root of 0 is 0.
1
E X A M P L E
Finding Real Roots
Find all real roots.
A
fourth roots of 81
= 81
4
A positive number has two real fourth roots. Because 3
(
)
4
-3
= 81, the roots are 3 and -3.
and
cube roots of -125
B
(
)
-5
3
= -125, the
A negative number has one real cube root. Because
root is -5.
sixth roots of -729
C
A negative number has no real sixth roots.
Find all real roots.
1a. fourth roots of -256 1b. sixth roots of 1 1c. cube roots of 125
610
610
Chapter 8 Rational and Radical Functions

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