DEFINITION OF Nth ROOT

If n is any positive integer, then the principal nth root of x is defined as follows :

√x = y  means x = yⁿ

If n is even, we must have a ≥ 0 and b ≥ 0.

Thus

⁴√81 = 3, because 3⁴ = 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 

√(3²) = √9 = 3

but

√(-3)² = √9 = 3 = |-3|

So, the equation √a² = a is not always true: it is true only when a ≥ 0. However, we can always write √a² = |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.

Properties of nth Roots

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√(aⁿ) = a, if n is odd

Example :

³√(-5)³ = -5

³√(-2)³ = -2

Property 5 :

ⁿ√(aⁿ) = |a|, if n is even

Example :

⁴√(-3)⁴ = |-3| = 3

Simplifying Expressions Involving nth Roots

Example 1 :

Simplify :

³√(x⁴)

Solution :

³√(x⁴) = ³√(x³x)

= ³√(x³) ⋅ √x

= ³√(x³) ⋅ ³√x

= x(³√x)

Example 2 :

Simplify :

⁴√(81x⁸y⁴)

Solution :

⁴√(81x⁸y⁴) = ⁴√81 ⋅ ⁴√x⁸ ⋅ ⁴√y⁴

= ⁴√(3⁴) ⋅ ⁴√(x²)⁴ ⋅ ⁴√y⁴

= 3x² ⋅ |y|

= 3x²y

Combining Radicals

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) - √(b³)

Solution :

√(25b) - √(b³) = √25√b - √(b² ⋅ b)

= √25√b - √(b²)√b

= √25√b - √(b²)√b

= 5√b - b√b

= (5 - b)√b

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