Download Section 8.1 Radical Expressions

11/13/2012
Section 8.1
Radical Expressions
radical
radical sign
root index
n
a
radicand
Find roots of numbers.
The opposite (or inverse) of squaring a number is taking its square root.
36 = 6, because 62 = 36.
We now extend our discussion of roots to include cube roots 3
,
fourth roots and higher roots.
4
,
n
The nth root of a, written
is,,
n
n
a
a , is a number whose nth power equals a. That
a = b means b n = a.
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When taking the square root, we do not express the 2
root index 2; we simply write rather than .
When taking higher roots the root index must be expressed. 3
8 = 2 since 23 = 8
CLASSROOM
EXAMPLE 1
Simplifying Higher Roots
Simplify.
Solution:
3
27
= 3, because 33 = 27
3
216
= 6, because 63 = 216
4
256
= 4, because 44 = 256
5
243
= 3, because 35 = 243
16
81
2
⎛ 2 ⎞ 16
= , because ⎜ ⎟ =
3
⎝ 3 ⎠ 81
0.064
= 0.4, because 0.43 = 0.064
4
4
3
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Finding Principal Roots
Even though 4 has two square roots, ‐2 and 2, the symbol means the principal square root or 4
positive square root, which in this case is 2.
Rules for nth Root:
If n is even and a is positive or 0, then n
a represents the principal nth root of a, and
‐ n a represents the negative nth root of a. If i
If n is even
and a is negative, then is not a real d i
i
h na i
l
number.
If n is odd, then there is exactly one nth root of a, n
written .
a
CLASSROOM
EXAMPLE 2
Finding Roots
Find each root. Solution:
36
=6
− 36
= −6
4
16
=2
− 4 16
= −2
4
−16
Not a real number.
5
=3
5
−243 = −3
243
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Find each root that is a real number. Use a calculator as necessary.
- 121
3
343
3
-125
3
-1000
4
625
4
-81
4