Download Algebra 2 Lesson 7

Roots and Radical Expressions
ALGEBRA 2 LESSON 7-1
Algebra 2
Lesson 7-1
(Page 363)
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Roots and Radical Expressions
ALGEBRA 2 LESSON 7-1
To simplify nth roots.
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Roots and Radical Expressions
ALGEBRA 2 LESSON 7-1
For any real numbers a and b,
and any positive integer n, if
a b
n
Then a is an nth root of b.
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Roots and Radical Expressions
ALGEBRA 2 LESSON 7-1
Type of Number
Number of Real
nth Roots when
n is even.
Number of Real
nth Roots when
n is odd.
Positive
2
1
Zero
1
1
Negative
None
1
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Roots and Radical Expressions
ALGEBRA 2 LESSON 7-1
Find all the real roots.
1
a. the cube root of 0.027, –125, and 64
Since 0.33 = 0.027, 0.3 is the cube root of 0.027.
Since (–5)3 = –125, –5 is the cube root of –125.
Since
1
4
3
=
1
1
1
,
is the cube root of
.
64 4
64
81
b. the fourth roots of 625, –0.0016, and 625
Since 54 = 625 and (–5)4 = 625, 5 and –5 are fourth roots of 625.
There is no real number with a fourth power of –0.0016.
3
Since 5
4
81
–3
= 625 and 5
4
81
3
3
81
= 625 , 5 and – 5 are fourth roots of 625 .
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Roots and Radical Expressions
ALGEBRA 2 LESSON 7-1
Find all the real roots.
a.
Find all the real fifth roots of 0, -1, and 32.
a. 0; no real fifth root; 2
b. 0; –1; 2 and –2
c. 0; –1; 2
d. 0; –1; no real fifth root
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Roots and Radical Expressions
ALGEBRA 2 LESSON 7-1
Find all the real roots.
b.
Find all the real square roots of 0.0001, -1, and
a. 0.01 and –0.01; no real square root;
b. 0.01; –1;
c. 0.01; no real square root;
d. 0.01; no real square root;
and
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36
121
and
Roots and Radical Expressions
ALGEBRA 2 LESSON 7-1
Find each real-number root.
a.
3
–1000
=
3
(–10)3
= –10
b.
Rewrite –1000 as the third power of a number.
Definition of nth root when n = 3, there is only one
real cube root.
–81
There is no real number whose square is –81.
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Roots and Radical Expressions
ALGEBRA 2 LESSON 7-1
Find each real-number root.
a. 3 27
a. no real-number root
b. –3
c. 3
d. 9
b. 4 81
a. no real-number root
b. –3
c. 3
d. 9
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c. 49
a. no real-number root
b. 7
c. 24.5
d. –7
Roots and Radical Expressions
ALGEBRA 2 LESSON 7-1
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Roots and Radical Expressions
ALGEBRA 2 LESSON 7-1
When a number has two real roots,
The positive root is called the
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Roots and Radical Expressions
ALGEBRA 2 LESSON 7-1
For any negative real number a ,
n
a a
n
When n is even.
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Roots and Radical Expressions
ALGEBRA 2 LESSON 7-1
Simplify each radical expression.
9x10
a.
9x10 =
32(x5)2 =
(3x5)2 = 3| x5|
Absolute value symbols ensure that the root is positive when x is negative.
b.
3
a3b3
3
(ab)3 = ab
Absolute value symbols must not be used here. If a or b is negative, then
the radicand is negative and the root must also be negative.
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Roots and Radical Expressions
ALGEBRA 2 LESSON 7-1
(continued)
c.
4
x16y4
4 x16y4
=
4 (x4)4(y)4
= x4 |y|
Absolute value symbols ensure that root is positive when y is negative.
They are not needed for x because x2 is never negative.
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Roots and Radical Expressions
ALGEBRA 2 LESSON 7-1
a. 4 x 2 y 4
a. –2 | x | y 2
b. 2 | x | y 2
c. | x | y 2
d. 2xy 2
b. 3 27c6
a. 3c 2
b. 3c 3
c. 3| c 3 |
d. –3c 2
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c. 4 x8 y12
a. x 2 | y 3 |
b. x2y8
c. x 2 y 3
d. x 2 | y 8 |