Download 5.5 Roots of Real Numbers

5.5 Roots of Real Numbers
Objectives:
1. Simplify radicals
2. Use a calculator to approximate
radicals.
Vocabulary
• Square root – For any real numbers a and b, if
a²=b, then a is a square root of b.
• Squaring a number and taking the square root
are inverse relations.
• nth root – For any real numbers a and b, and
n
any positive integer n, if a =b, then a is an nth
root of b. Example: 2⁴=16, so 2 is the 4th root
of 16.
• Some common roots to remember:
4
x x
4
5
x x
5
Roots
Radical sign
index
4
• Parts of a root
81
radicand
• Principal root – the nonnegative root
•
means the principal square root of x.
x No index is given, so it is understood
to be 2.
•  x means the opposite of the principal
square root
•  x indicates both square roots of x.
Real nth roots of b,
n
n
n
even
b
if b > 0
or  n b
b
n
b
if b < 0
one positive root,
one negative root
No real roots
 81  3
is not a real
number
One positive root,
no negative roots
No positive roots,
one negative root
16
4
odd
b=0
3
64  4
5
1024  4
One real
root, 0
n
0 0
nth roots of an even power
• When you take the nth root of a even power
and the result is an odd power, you must take
the absolute value of the result to ensure the
answer is non-negative.
8
12
x  x
8
x
x  x
4
2
4
6
x
4
Examples
Use a calculator to approximate each value to
three decimal places.
 17.029
3
589  8.382
290
6
681  2.966
Examples
Simplify
400
4
.
 20
1296
6
3
64  4
4
1
1

3
81
 a
5
5
 a
100x y  10x y
4
3
2
8
125x y  5
4
3
x y
2
( x  4)  x  4
9
x
2
3
2

(
x

3)
 6x  9
 x 3
Homework
page 248
16-56 even