The cube root of a number is denoted by ∛
The cube root of a number x is that number whose cube gives x. We denote the cube root of x by ∛x
Thus, 3√64 = cube root of 64 = 3∛4 × 4 × 4 = ∛4³ = 4
For example:
(i) Since (2 × 2 × 2) = 8, we have ∛8 = 2
(ii) Since (5 × 5 × 5) = 125, we have ∛125 = 5
To find the cube root of a given number, proceed as follows:
Step I. Express the given number as the product of primes.
Step II. Make groups in triplets of the same prime.
Step III. Find the product of primes, choosing one from each triplet.
Step IV. This product is the required cube root of the given number.
Note: If the group in triplets of the same prime factors cannot complete, then the exact cube root cannot be found.
1. Evaluate the cube root: ∛216
Solution:
By prime factorization, we have
216 = 2 × 2 × 2 × 3 × 3 × 3
= (2 × 2 × 2) × (3 × 3 × 3)
Therefore, ∛216 = (2 × 3) = 6
2. Evaluate the cube root: ∛343
Solution:
By prime factorization, we have
343 = 7 × 7 × 7
= (7 × 7 × 7).
Therefore, ∛343 = 7
3. Evaluate the cube root: ∛2744
Solution:
By prime factorization, we have
2744 = 2 × 2 × 2 × 7 × 7 × 7
= (2 × 2 × 2) × (7 × 7 × 7).
Therefore, ∛2744 = (2 × 7) = 14
Let (a) be a positive integer. Then, (-a) is a negative integer.
We know that (-a)³ = -a³.
Therefore, ∛-a³ = -a.
Thus, cube root of (-a³) = -(cube root of a³).
Thus, = ∛-x = - ∛x
For example:
Find the cube root of (-1000).
Solution:
We know that ∛-1000 = -∛1000
Resolving 1000 into prime factors, we get
1000 = 2 × 2 × 2 × 5 × 5 × 5
= (2 × 2 × 2) × (5 × 5 × 5)
Therefore, ∛1000 = (2 × 5) = 10
Therefore, ∛-1000 = -(∛1000) = -10
We have, ∛ab = (∛a × ∛b).
For example:
1. Evaluate: ∛(125 × 64).
Solution:
(∛125 × 64)
= ∛125 × ∛64
= [∛{5 × 5 × 5}] × [∛{4 × 4 × 4}]
= (5 × 4)
= 20
2. Evaluate: ∛(27 × 64).
Solution:
(∛27 × 64)
= ∛27 × ∛64
= [∛{3 × 3 × 3}] × [∛{4 × 4 × 4}]
= (3 × 4)
= 12
3. Evaluate: ∛[216 × (-343)].
Solution:
∛[216 × (-343)]
= ∛216 × ∛-343
= [∛{6 × 6 × 6}] × [∛{(-7) × (-7) × (-7)}]
= [6 × (-7)]
= -42.
We define: ∛(a/b) = (∛a)/(∛b)
For example:
Evaluate:
{∛(216/2197)
Solution:
∛(216/2197)
= ∛216/∛2197
= [∛(6 × 6 × 6)]/[ ∛(13 × 13 × 13)]
= 6/13
Cube root of a fraction is a fraction obtained by taking the cube roots of the numerator and the denominator separately.
If a and b are two natural numbers, then ∛(a/b) = (∛a)/(∛b)
For Example:
∛(-125/512)
= ∛(-125)/∛512
= ∛{(-5) × (-5) × (-5)}/∛{8 × 8 × 8}
= -5/8
Express the given decimal in the fraction form and then find the cube root of the numerator and denominator separately and convert the same into decimal.
For Example:
Find the cube root of 5.832.
Solution:
Converting 5.832 into fraction, we get 5832/1000
Now ∛5832/1000 = ∛5832/∛1000
= ∛(2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3)/∛(2 × 2 × 2 × 5 × 5 × 5)
= 2 × 3 × 3/2 × 5
= 18/10
= 1.8
● Cube and Cube Roots
To Find if the Given Number is a Perfect Cube
Method for Finding the Cube of a Two-Digit Number
● Cube and Cube Roots - Worksheets
Worksheet on Cube and Cube Root
8th Grade Math Practice
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