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Section 8.1 Simplifying Radical Expressions
A radical expression is one that contains roots. The number under the radical sign is called the radicand.
All positive, real numbers have roots, but negative numbers do not. The perfect squares such as 4, 9, 16, 25, 36... all have roots that are whole numbers:
and all other positive numbers in between such as 2, 3, 5, 6, 7, 8, 10, 11... also have roots, but they are irra-tional numbers:
= 1.4142...
= 1.7320...
= 2.2360...
= x
Negative numbers do not have roots. A negative root, multiplied by its identity, another negative root, produces a positive number (–2 –2 = +4). Thus, the square root of positive numbers can be both, posi-tive and negative, leaving negative numbers without square roots.
Example: If 3 3 = 9 and –3 –3 = 9
then the square root of 9 can be both +3 and –3
and the square root of , for example, cannot be found.
Practice:Simplify.
81
4 2=
9 3=
16 4=
25 5=
36 6=
2
3
5
x2
y4
y2=
9 3=
9–
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3.
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5.
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49
81
144
289
324
121
22–
39
x4
120
250
18
800
340
90–
y6
500
900
9y8
16x10
132 Chapter 8: Radicals
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RADICAL EXPRESSIONS
A radical expression, also called a radicand, is any expression found under a radical ( ). Moreover, neg-ative outcomes of radical expressions are not real numbers.
Example: What values of x will make a real number?
When x = 0, 1, 2 , 3 or 4, the radical is not a real number because the result is a negative radicand. How-ever, x values of 5 or above are possible.
When x = 0 the radical is no answer because the radicand is negative
When x = 1 the radical is no answer because the radicand is negative ...
When x = 4 the radical is no answer because the radicand is negative
When x = 5 the radical is = 0 (first real number)
Answer: x 5
Example: What values will make a real number?
Because squaring a negative number gives always a positive answer, in the example above all real numbers
(including all negative numbers) are values that would make a real number.
Example: What values will make a real number?
(x could be any real number)
Example: What values will make a real number?
5x + 7 0 5x –7
Answer: x
Example: What values will make a real number?
= (x could be any real number)
x
x 5–
5–
4–
1–
0
x2
2+
x2
2+
x2
x2 x=
5x 7+
x 75---–
75---–
x 3– 2
x 3– 2x 3–
8.1 Simplifying Radical Expressions 133
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Example: What values will make a real number?
(y could be any real number)
Example: What values will make a real number?
Factor first: =
= (x and y could be any real number)
Practice:Find the value of the variable that would yield a real number for the expression.
Simplify.
SIMPLIFICATION OF RADICALS (for nonnegative real numbers)To simplify a radical means to reduce the radical to the point of not having any perfect squares represented
in the radicand. In other words, every two of the same number—or base—under the radical sign, repre-sents one outside the radical sign, the rest stays under the radical.
Example: Simplify
Use prime factorization to simplify radical = (simplify 32)
Answer:
19---y
2
19---y
2 13--- y=
4x2
12xy– 9y2+
4x2
12xy– 9y2+ 2x 3y– 2x 3y–
2x 3y– 2x 3y– 2x 3y–
18
18 2 3 3
3 2
1.
2.
3.
4.
5.
6.
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11.
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x 8–
x 5+
2x 9–
7x
x 18–
x2
12+
3x 4–
x2 6+
x 1–
x 20+
y2 4–
5z 12–
13.
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27.
y4
x 8+ 2
1625------y
2
x y– 2
x2
12xy– 36y2+
16x2 40xy– 25+
a– 2
814------y
2
x2
2xy y2+ +
16b– 2
x 4+ 2
y2 9– 9
------------------2
25x2
90xy 81+ +
49b– 2
9 a 12– 4
134 Chapter 8: Radicals
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Example: Simplify
Use prime factorization to simplify radical = = 4b
Perfect square, radical sign gone
Example: Simplify
Use prime factorization to simplify radical = = 5
Example: Simplify
Use prime factorization to simplify radical = =
Example: Simplify
Use prime factorization to simplify radical = =
Example: Simplify
Factor the 3 first
Simplify the perfect trinomial square =
Practice:Simplify. Assume all variables to be nonnegative.
16b2
16b2
4 4 b b
25b
25b 5 5 b b
125a3
125a3
5 5 5 a a a 5a 5a
y7
y7
yyyyyyy y3
y
3a2
30a 75+ +
3 a2
10a 25+ +
3 a 5+ a 5+ a 5+ 3
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12
27x2
32z
98a5
a4b
3c
18x2
60x– 50+
75
60y4
250x2y
3z
4
a b+ 2
m4n
7
4a2
56a 196+ +
200
48g5h3
72b5c
3
36 x y+ 3
45a3b
2c
8
20x2
60x 45+ +
80
64wy5
xy9
x 3– 3
2x 5+ 7
a 8+ 4
4x2
12x– 9+
a b+ 5
6x 17– 3
x 8– 2
9x2 30xy– 25+
s t u–+ 6
x 1+ 13
8x 3+ 7
2a 9+ 3
x2
4x 4+ +
x5y
8
5x 12– 4
x 2+ 8
9x2 6xy 1+ +
y2 4– 25
------------------3
4x2
8x– 4+
4 b– 7
8.1 Simplifying Radical Expressions 135
