In general, for an integer n greater than 1, if b^{n} = 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.
If n is an even number
If a < 0 If a = 0 If a > 0 |
No real n^{th} roots One real n^{th} root ^{n}√0 = 0 Two real n^{th} roots ±^{n}√a = ±a^{1/n} |
If n is an odd number
If a < 0 If a = 0 If a > 0 |
One real n^{th} root ^{n}√a = a^{1/n} One real n^{th} root ^{n}√0 = 0 One real n^{th} roots ^{n}√a = a^{1/n} |
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
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM