Cube Root

# Cube Root

Table of Contents

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.

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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.

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