Roots of real numbers and radical expressions
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5.5 Roots of Real Numbers and Radical
Expressions
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Definition of nth Root
** For a square root the value of n is 2.
For any real numbers a and b and any positive integers n, if an = b, then a is the nth root of b.
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Definitions
A perfect square is the square of a natural number. 1, 4, 9, 16, 25, and 36 are the first six perfect squares.
A perfect cube is the cube of a natural number. 1, 8, 27, 64, 125, and 216 are the first six perfect cubes.
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Notation
814index
radical
radicand
Note: An index of 2 is understood but not written in a square root sign.
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Simplifying Radicals
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Simplify 814
To simplify means to find x in the equation:
x4 = 81
Solution: = 3 814
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Principal Root
The nonnegative root of a number
64
64
64
Principal square root
Opposite of principal square root
Both square roots
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Summary of Roots
even
odd
one + root one - rootone + root no - roots
no real roots
no + roots one - root
one real root, 0
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Examples
4
4
1. 169
2. - 8 -3
x
x
2213 x 213x
228 3 x
28 3x
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Examples
323 5 x3 6
3 3 3
3. 125
4.
x
m n
25 x
33 mn mn
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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.
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Examples
44
626
1.
2.
an
xy
EvenEven
EvenEven
an
Odd
2 x yOdd
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6
1826
3.
4. 3
x
yEven
Even
3 x
Odd
Odd
Even
Even
2
323- y
323- y