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Roots of Real Numbers and Radical
Expressions
Definition of Square Root
For any real numbers a and b, if , then a is a square root of b.Ex: , 5 is a square root of 25
ba 2
2552
Definition of nth Root• For an real numbers a and b, and any positive n,
if , then a is an nth root of b.
• EX: Since , 2 is a fifth root of 32.
ban
3225
Notation
814indexradical
radicand
Note: An index of 2 is understood but not written in a square root sign.
Simplify 814
To simplify means to find x in the equation:
x4 = 81
Solution: = 3 814
Principal Root
The nonnegative root of a number
64 64 64
Principal square root
Opposite of principal square root Both square roots
Summary of Roots
even
odd
one + root one - rootone + root no - roots
no real roots
no + roots one - root
one real root, 0
Examples
4
4
1. 169
2. - 8 -3
x
x
2213 x 213x
228 3 x
28 3x
Examples
323 5 x3 6
3 3 3
3. 125
4.
x
m n
25 x
33 mn mn
Taking nth roots of variable expressions
Using absolute value signs
If the index (n) of the radical is even, the power under the radical sign is even, and the resulting power is odd, then we must use an absolute value sign.
Examples
44
626
1.
2.
an
xy
EvenEven
EvenEven
an
Odd
2 x yOdd
6
1826
3.
4. 3
x
yEvenEven
3 x
Odd
Odd
Even
Even
2
323- y
323- y