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Chapter 0 Section 6
Rational Exponents and Radicals
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Radicals and Radical Notation
n bThe n is called the “index”—it indicates what root to find.
The b is called the radicand– it indicates the number you are trying to find the root of…
With radical notation, if n=2 we normally don’t put the 2 in for the index, we just know it as the “square root”. If n=3 we call it the “cube root”…but we must put the 3 in for the index.
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Evaluating Square Roots
121
64
25
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Evaluating Cube Roots
3 64
3 125
3 8
3 27
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Summary of evaluating
even #
odd #
NOT a real number
IS a real number
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Mixed Examples
49 5 32 81
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Radicals as Fractional Exponents
There is a relationship between radical notation and exponents:
nn bb1
The index becomes a fractional exponent instead.
All of the properties of radicals work for fractional exponents too.
The denominator of a fractional exponent is the index.
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Convert to radical form and evaluate
31
64 32
125 23
363 64
4
23 125
25
25
336
36
216
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Properties of Radicals
nnn baab
n
n
n
b
a
b
a
Product property: The root of a product is the product of the roots
Quotient property: The root of a quotient is the quotient of the roots
bnn b aa )( Power property: The root of a power is the power of the root
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Simplify the square roots
50 42108 yx 52340 cba
225
25
42336 yx
36 2xy
ccaba 422104
acabc 102 2
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Simplifying algebraic expressions
64y63 643 y63= = 4y2
m4
n8
4=
m44
n84=mn2
*You want the exponents of the variables to be a multiple of the index.
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More simplifying algebraic expressions
If the exponents on the variables are not multiples of the index, rewrite as a product so one of the exponents is a multiple of the index.
4 10844 cba 4 28844 ccba
4 24 884 4ccba
4 222 4ccab
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Simplify the expressionsSimplest Radical Form…
1353 = 273 5
= 273 53
533=
xx 612 272x
2236 x2236 x
26x
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6. 274 34
Evaluate the expressions
123 183 12 183= 2163= = 6
804
54
805
4= = 164 = 2
3SOLUTION SOLUTION
7.
232503
5
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Numeric expressions with +/-Combining Like Radicals…just like with combining like variables, you may combine radical expressions if the index and radicand match.
233 23–=543 – 23 = 23273 23– 23(3 – 1)= = 2 23
If the index and radicands don’t match after you have simplified, you may not combine the radicals and radicands… they aren’t like.
139133 1393 1312
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Add or Subtract
7157873
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Rationalizing the Denominator
7
1
If a reduced fraction still has a radical in the denominator it is not really reduced…there is still some work to do…you need to “rationalize the denominator”, that means make the denominator a Rational number instead of an Irrational number.
Think: what times the square root of 7 gives 7…
7
1
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5
1
5
1
5
5
5
1
5
5
103
2
10
10
103
2
103
102
15
10
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Conjugates
If the denominator of the fraction also has an addition or subtraction along with the radical you need to use something called the “conjugate”…some examples of conjugate pairs are listed below.
115,115
72,72
53,53
Multiply each pair together (use the same rules for FOIL) and see if you notice something special…
When multiplying conjugates, the O and I steps can be skipped (they cancel each other out)
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Simplify
62
2
Find the conjugate and multiply numerator and denominator by that conjugate.
62
62
62
2
2
624
62
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Simplify
23
62
23
23
29
1263226
7
3263226
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Properties
nm
a mna1
mn a
n ma
naa n
1
11
nma
a nm 1