FIND THE INDICATED NTH ROOTS OF A

In general, for an integer n greater than 1, if bⁿ = a, then b is an nth root of a. An n^th root of a is written as ⁿ√a , where n is the index of the radical.

Let n be an integer (n > 1) and let a be a real number.

Find the indicated real n^th root(s) of a.

Problem 1 :

If n = 3, a = −216

Solution :

We can write the given details as cube root of -216. That is

= ∛-216

= ∛-6 ⋅ (-6) ⋅ (-6)

= -6

Problem 2 :

n = 4, a = 81

Solution :

Here n is even. So, we will have two solutions.

We can write the given details as fourth root of 81. That is,

= ∜81

= ∜(3 ⋅ 3 ⋅ 3 ⋅ 3)

= ±3

Problem 3 :

n = 4, a = 16

Solution :

Here n is even. So, we will have two solutions.

We can write the given details as fourth root of 81. That is,

= ∜16

= ∜(2 ⋅ 2 ⋅ 2 ⋅ 2)

= ±2

Problem 4 :

n = 2, a = −49

Solution :

Here n is even and value of a is less than 0. So, it will have no real roots.

Problem 5 :

n = 3, a = −125

Solution :

Here n is odd and value of a is less than 0. 

= ∛-125

= ∛-5 ⋅ (-5) ⋅ (-5)

= -5

Problem 6 :

n = 5, a = 243

Solution :

Here n is odd and value of a > 0. 

= 5^th root (243)

= 5^th root (3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3)

= 3