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.

Solved Examples of Cube Root using step by step with explanation

**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

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** **From Cube Root to HOME PAGE**

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