7­1 Roots and Radical Expressions 2011
March 02, 2011
7­1 Roots and Radical Expressions
Objective:
Simplify nth roots.
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7­1 Roots and Radical Expressions 2011
March 02, 2011
Check Skills You'll Need
Write each number as a square of a number.
1. 25
2. 0.09
3. 4/49
Write each expression as a square of an expression.
4. x10
5. x4y2
6. 169x6y12
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7­1 Roots and Radical Expressions 2011
March 02, 2011
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7­1 Roots and Radical Expressions 2011
March 02, 2011
List the squares of all the numbers 1‐15.
1­
2­
3­
4­
5­
6­
7­
8­
9­
10­
11­
12­
13­
14­
15­
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7­1 Roots and Radical Expressions 2011
March 02, 2011
List the cubes of all the numbers 1‐10.
1­
2­
3­
4­
5­
6­
7­
8­
9­
10­
5
7­1 Roots and Radical Expressions 2011
March 02, 2011
Look at the following pattern:
Since 52 = 25, 5 is a square root of 25.
Since 53 = 125, 5 is a cube root of 125.
Since 54 = 625, 5 is a fourth root of 625.
Since 55 = 3125, 5 is a fifth root of 3125.
This pattern leads to the definition of nth root.
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7­1 Roots and Radical Expressions 2011
March 02, 2011
Since 22 = 4 and (‐2)2 = 4, both 2 and ‐2 are fourth roots of 4.
Since there is no real number x such that x2 = ‐4,
‐4 has no real fourth root.
Since ‐5 is the only real number whose cube is ‐125,
‐5 is the only real cube root of ‐125.
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7­1 Roots and Radical Expressions 2011
March 02, 2011
Find all the real cube roots.
a. 0.008
(0.2)3= 0.008
b. ‐1000
(‐10)3 = ‐1000
c. 1/27
(1/3)3 = 1/27
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7­1 Roots and Radical Expressions 2011
March 02, 2011
Find all the real fifth roots.
d. 0
e. ‐1
f. 32
0 ‐1 2
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7­1 Roots and Radical Expressions 2011
March 02, 2011
A radical sign is used to indicate a root.
The index gives the degree of the root.
The number under the radical sign is the radicand.
When a number has TWO real roots, the
POSITIVE root is called the PRINCIPAL ROOT.
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7­1 Roots and Radical Expressions 2011
March 02, 2011
When a number has TWO real roots, the
POSITIVE root is called the PRINCIPAL ROOT.
The PRINCIPAL fourth root of 16 is 2
4
because equals . The other
∜16
∜2
fourth root of 16 is written as ­ , ∜16
which equals ­2.
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7­1 Roots and Radical Expressions 2011
March 02, 2011
Find each real number root.
a. ∛-8
b. √-100
c. ∜81
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7­1 Roots and Radical Expressions 2011
March 02, 2011
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7­1 Roots and Radical Expressions 2011
March 02, 2011
Simplify each radical expression.
a. √4x6
b. ∛a3b6
c. ∜x4y8
√22(x3)2
∛a3(b2)3
∜x4(y2)4
√(2x3)2
∛(ab2)3
∜(xy2)4
2|x3|
ab2
|x|y2
Absolute value symbols ensure that the root is positive when x is negative.
Absolute value symbols must not be used here. If a is negative, then the radicand is negative and the root must also be negative.
Absolute value symbols ensure that the root is positive when x is negative. They are not needed for y because y2 is never negative.
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7­1 Roots and Radical Expressions 2011
March 02, 2011
HOMEWORK
pg. 372 #1 ­ 28,
39 ­ 54
&
pg. 368 (1­24) 15
7­1 Roots and Radical Expressions 2011
March 02, 2011
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