Cube Root of a number can be obtained by doing the inverse operation of calculating cube. In general terms, the cube root of a number is identified by a number that multiplied by itself thrice gives you the cube root of that number. The cube root of any number is denoted with the symbol ∛. Look at the Cube and Cube Roots solved examples and their explanations to learn them easily.
For example, the cube root of a number x is represented as ∛x.
How to Find the Cube Root of a Number?
Simply, Note down the product of primes a number. Then, form the groups in triplets using the product of primes a number. After that take one number from each triplet. The selected single number is the required cube root of the given number.
Note: If you find a group of prime factors that cannot form a group in triplets they remain the same and their cube root cannot be found.
Cube Root of a Number Solved Examples
(i) Find the Cube Root of a number 64?
Answer:
Write the product of primes of a given number 64 those form groups in triplets.
Cube Root of 64 = ∛64 = ∛(4 × 4 × 4)
Take one number from a group of triplets to find the cube root of 64.
Therefore, 4 is the cube root of a given number 64.
4 is the cube root of a given number 64.
(ii) Find the Cube Root of a number 8?
Answer:
Write the product of primes of a given number 8 those form groups in triplets.
Cube Root of 8= ∛8= ∛(2 × 2 × 2)
Take one number from a group of triplets to find the cube root of 8.
Therefore, 2 is the cube root of a given number 8.
2 is the cube root of a given number 8.
(iii) Find the Cube Root of a number 125?
Answer:
Write the product of primes of a given number 125 those form groups in triplets.
Cube Root of 125= ∛125= ∛(5 × 5 × 5)
Take one number from a group of triplets to find the cube root of 125.
Therefore, 5 is the cube root of a given number 125.
5 is the cube root of a given number 125.
(iv) Find the Cube Root of a number 27?
Answer:
Write the product of primes of a given number 27 those form groups in triplets.
Cube Root of 27 = ∛27 = ∛(3 × 3 × 3)
Take one number from a group of triplets to find the cube root of 27.
Therefore, 3 is the cube root of a given number 27.
3 is the cube root of a given number 27.
(iv) Find the Cube Root of a number 216?
Answer:
Write the product of primes of a given number 216 those form groups in triplets.
Cube Root of 216 = ∛216 = ∛(6 × 6 × 6)
Take one number from a group of triplets to find the cube root of 216.
Therefore, 6 is the cube root of a given number 216.
6 is the cube root of a given number 216.
Finding Cube Root by Prime Factorisation Method
Find the cube root of a number using the Prime Factorisation Method with the help of the below steps.
Step 1: Firstly, take the given number.
Step 2: Find the prime factors of the given number.
Step 3: Group the prime factors into each triplet.
Step 4: Collect each one factor from each group.
Step 5: Finally, find the product of each one factor from each group.
Step 6: The resultant is the cube root of a given number.
Cube Root of a Number by Prime Factorisation Method Solved Examples
(i) Find the Cube Root of 216 by Prime Factorisation Method?
Answer:
Firstly, find the prime factors of the given number.
216 = 2 × 2 × 2 × 3 × 3 × 3
Group the prime factors into each triplet.
216 = (2 × 2 × 2) × (3 × 3 × 3)
Collect each one factor from each group.
2 and 3
Finally, find the product of each one factor from each group.
∛216 = 2 × 3 = 6
6 is the cube root of 216.
(ii) Find the Cube Root of 343 by Prime Factorisation Method?
Answer:
Firstly, find the prime factors of the given number.
343 = 7 × 7 × 7
Group the prime factors into each triplet.
343 = (7 × 7 × 7)
Collect each one factor from each group.
7
Finally, find the product of each one factor from each group.
∛343 = 7
7 is the cube root of 343.
(iii) Find the Cube Root of 2744 by Prime Factorisation Method?
Answer:
Firstly, find the prime factors of the given number.
2744 = 2 × 2 × 2 × 7 × 7 × 7
Group the prime factors into each triplet.
2744 = (2 × 2 × 2) × (7 × 7 × 7).
Collect each one factor from each group.
2 and 7
Finally, find the product of each one factor from each group.
∛2744 = 2 × 7 = 14
14 is the cube root of 2744.
Cube Roots of Negative Numbers
Cube Root of a negative number is always negative. If -m be a negative number. Then, (-m)³ = -m³.
Therefore, ∛-m³ = -m.
cube root of (-m³) = -(cube root of m³).
∛-m = – ∛m
Solved Examples of Cube Root of a Negative Numbers
(i) Find the Cube Root of (-1000)
Answer:
Firstly, find the prime factors of the number 1000.
1000 = 2 × 2 × 2 × 5 × 5 × 5
Group the prime factors into each triplet.
1000 = (2 × 2 × 2) × (5 × 5 × 5).
Collect each one factor from each group.
2 and 5
Finally, find the product of each one factor from each group.
∛1000 = 2 × 5 = 10
∛-m = – ∛m
∛-1000 = – ∛1000 = -10
-10 is the cube root of (-1000).
(ii) Find the Cube Root of (-216)
Answer:
Firstly, find the prime factors of the number 216.
216 = 2 × 2 × 2 × 3 × 3 × 3
Group the prime factors into each triplet.
216 = (2 × 2 × 2) × (3 × 3 × 3)
Collect each one factor from each group.
2 and 3
Finally, find the product of each one factor from each group.
∛216 = 2 × 3 = 6
∛-216 = – ∛216
∛-216= – ∛216= -6
-6 is the cube root of -216.
How to Find Cube Root of Product of Integers?
Cube Root of Product of Integers can be solved by using ∛ab = (∛a × ∛b)
Solved Examples:
(i) Find ∛(125 × 64)?
Answer:
Firstly, apply the cube root to both integers.
∛ab = (∛a × ∛b)
∛(125 × 64) = ∛125 × ∛64
Then, find the prime factors for each integer separately.
[∛{5 × 5 × 5}] × [∛{4 × 4 × 4}]
Take each integer from the group in triplets and multiply them to get the cube root of a given number.
(5 × 4) = 20
20 is the cube root of ∛(125 × 64).
(ii) Find ∛(27 × 64)?
Answer:
Firstly, apply the cube root to both integers.
∛ab = (∛a × ∛b)
∛(27 × 64) = ∛27 × ∛64
Then, find the prime factors for each integer separately.
[∛{3 × 3 × 3}] × [∛{4 × 4 × 4}]
Take each integer from the group in triplets and multiply them to get the cube root of a given number.
(3 × 4) = 12
12 is the cube root of ∛(27 × 64).
(iii) Find ∛[216 × (-343)]?
Answer:
Firstly, apply the cube root to both integers.
∛ab = (∛a × ∛b)
∛[216 × (-343)] = ∛216 × ∛-343
Then, find the prime factors for each integer separately.
[∛{6 × 6 × 6}] × [∛{(-7) × (-7) × (-7)}]
Take each integer from the group in triplets and multiply them to get the cube root of a given number.
[6 × (-7)] = -42
-42 is the cube root of ∛[216 × (-343)].
Cube Root of a Rational Number
The Cube Root of a Rational Number can be calculated with the help of ∛(a/b) = (∛a)/(∛b). Apply the Cube Root separately to each integer available on the numerator and the denominator to find the cube root of a rational number.
Solved Examples of Cube Root of a Rational Number
(i) Find ∛(216/2197)
Answer:
Firstly, apply the cube root to both integers.
∛(a/b) = (∛a)/(∛b)
∛(216/2197) = ∛216/∛2197
Then, find the prime factors for each integer separately.
[∛(6 × 6 × 6)]/[ ∛(13 × 13 × 13)]
Take each integer from the group in triplets to get the cube root of a given number.
6/13
6/13 is the cube root of ∛(216/2197).
(ii) Find ∛(27/8)
Answer:
Firstly, apply the cube root to both integers.
∛(a/b) = (∛a)/(∛b)
∛(27/8) = ∛27/∛8.
Then, find the prime factors for each integer separately.
[∛(3 × 3 × 3)]/[ ∛(2 × 2 × 2)]
Take each integer from the group in triplets to get the cube root of a given number.
3/2
3/2 is the cube root of ∛(27/8).
How to Find the Cube Root of Decimals?
The Cube Root of Decimals can easily be solved by converting them into fractions. After converting the decimal number into a fraction apply the cube root to the numerator and denominator separately. Then, convert the resultant value to decimal.
Cube Root of Decimals Solved Examples
(i) Find the cube root of 5.832.
Answer:
Conver the given decimal 5.832 into a fraction.
5.832 = 5832/1000
Now, apply the cube root to the fraction.
∛5832/1000
Apply the cube root to both integers.
∛(a/b) = (∛a)/(∛b)
∛5832/1000 = ∛5832/∛1000.
Then, find the prime factors for each integer separately.
∛(2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3)/∛(2 × 2 × 2 × 5 × 5 × 5)
Take each integer from the group in triplets to get the cube root of a given number.
(2 × 3 × 3)/(2 × 5) = 18/10
Convert the fraction into a decimal
18/10 = 1.8
1.8 is the cube root of 5.832.