If n is any positive integer, then the principal nth root of x is defined as follows :
√x = y means x = y^{n}
If n is even, we must have a ≥ 0 and b ≥ 0.
Thus
^{4}√81 = 3, because 3^{4} = 81 and 3 ≥ 0
^{3}√(-8) = -2, because (-2)^{3} = -8
But √(-8), )^{4}√(-8) and ^{6}√(-8) are not defined.
For instance √(-8) is not defined, because the square of every real number is nonnegative.
Notice that
√(3^{2}) = √9 = 3
but
√(-3)^{2} = √9 = 3 = |-3|
So, the equation √a^{2 }= a is not always true: it is true only when a ≥ 0. However, we can always write √a^{2 }= |a|. This last equation is true not only for square roots, but for any even root.
This and other rules used in working with nth roots are listed below. In each property we assume that all the given roots exist.
Property 1 :
^{n}√(ab) = ^{n}√a ⋅ ^{n}√b
Example :
^{3}√(-8 ⋅ 27) = ^{3}√(-8) ⋅ ^{3}√27
= (-2)(3)
= -6
Property 2 :
^{n}√(a/b) = ^{n}√a/^{n}√b
Example :
^{4}√(16/81) = ^{4}√16/^{4}√81
= 2/3
Property 3 :
^{m}√[^{n}√a] = ^{mn}√a
Example :
√[^{3}√729] = ^{6}√729
= 3
Property 4 :
^{n}√(a^{n}) = a, if n is odd
Example :
^{3}√(-5)^{3} = -5
^{3}√(-2)^{3} = -2
Property 5 :
^{n}√(a^{n}) = |a|, if n is even
Example :
^{4}√(-3)^{4} = |-3| = 3
Example 1 :
Simplify :
^{3}√(x^{4})
Solution :
^{3}√(x^{4}) = ^{3}√(x^{3}x)
= ^{3}√(x^{3}) ⋅ √x
= ^{3}√(x^{3}) ⋅ ^{3}√x
= x(^{3}√x)
Example 2 :
Simplify :
^{4}√(81x^{8}y^{4})
Solution :
^{4}√(81x^{8}y^{4}) = ^{4}√81 ⋅ ^{4}√x^{8 }⋅ ^{4}√y^{4}
= ^{4}√(3^{4}) ⋅ ^{4}√(x^{2})^{4 }⋅ ^{4}√y^{4}
= 3x^{2 }⋅ |y|
= 3x^{2}y
Example 3 :
Simplify :
√32 + √200
Solution :
√32 + √200 = √(16 ⋅ 2) + √(100 ⋅ 2)
= √16 ⋅ √2 + √100 ⋅ √2
= 4√2 + 10√2
= (4 + 10)√2
= 14√2
Example 4 :
Simplify the following expression, if b > o.
√(25b) - √(b^{3})
Solution :
√(25b) - √(b^{3}) = √25√b - √(b^{2 }⋅ b)
= √25√b - √(b^{2})√b
= √25√b - √(b^{2})√b
= 5√b - b√b
= (5 - b)√b
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