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Find the concept-wise Cube and Cube Root Worksheets and begin your practice. Get various questions from Cube Root like finding the cube root of a number, cube root of a number using prime factorization method, cube root of negative numbers, cube root of rational numbers, etc. Get Step by Step Solutions for the Cube Root Problems and learn different topics involved easily.

1. Find the Cube Root of the given numbers?

(i) 512 (ii) 1728 (iii) 216 (iv) 1331

## Solution:

(i) 512

Write the product of primes of a given number 512 those form groups in triplets.

Cube Root of 512 = ∛512 = ∛(8 × 8 × 8)

Take one number from a group of triplets to find the cube root of 512.

Therefore, 8 is the cube root of a given number 512.

(ii) 1728

Write the product of primes of a given number 1728 those form groups in triplets.

Cube Root of 1728 = ∛1728 = ∛(12 × 12 × 12)

Take one number from a group of triplets to find the cube root of 1728.

Therefore, 12 is the cube root of a given number 1728.

(iii) 216

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.

(iv) 1331

Write the product of primes of a given number 1331 those form groups in triplets.

Cube Root of 1331 = ∛1331 = ∛(11 × 11 × 11)

Take one number from a group of triplets to find the cube root of 1331.

Therefore, 11 is the cube root of a given number 1331.

2. Find the Cube Root of a Number by Prime Factorisation Method?

(i) 15625 (ii) 3375 (iii) 216 (iv) 13824

## Solution:

(i) 15625

Firstly, find the prime factors of the given number.

15625 = 5 × 5 × 5 × 5 × 5 × 5

Group the prime factors into each triplet.

15625 = (5 × 5 × 5) × (5 × 5 × 5).

Collect each one factor from each group.

5 and 5

Finally, find the product of each one factor from each group.

∛15625= 5 × 5 = 25

25 is the cube root of 15625.

(ii) 3375

Firstly, find the prime factors of the given number.

3375 = 5 × 5 × 5 × 3 × 3 × 3

Group the prime factors into each triplet.

3375 = (5 × 5 × 5) × (3 × 3 × 3).

Collect each one factor from each group.

5 and 3

Finally, find the product of each one factor from each group.

∛3375= 5 × 3 = 15

15 is the cube root of 3375.

(iii) 216

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.

(iv) 13824

Firstly, find the prime factors of the given number.

13824 = 6 × 6 × 6 × 4 × 4 × 4

Group the prime factors into each triplet.

13824 = (6 × 6 × 6) × (4 × 4 × 4).

Collect each one factor from each group.

6 and 4

Finally, find the product of each one factor from each group.

∛13824 = 6 × 4 = 24

24 is the cube root of 13824.

3. Find the Cube Roots of Negative Numbers?

(i) (-27) (ii) (-1728) (iii) (-2744) (iv) (-512)

## Solution:

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

(i) (-27)

Find the prime factors of the number 27.

27 = 3 × 3 × 3

Group the prime factors into each triplet.

27 = (3 × 3 × 3)

Collect each one factor from each group.

3

Finally, find the product of each one factor from each group.

∛27= 3

∛-m = – ∛m

∛-27= – ∛27= -3

-3 is the cube root of (-27).

(ii) (-1728)

Find the prime factors of the number 1728.

1728 = 2 × 2 × 2 × 6 × 6 × 6

Group the prime factors into each triplet.

1728 = (2 × 2 × 2) × (6 × 6 × 6)

Collect each one factor from each group.

2 × 6

Finally, find the product of each one factor from each group.

∛1728= 12

∛-m = – ∛m

∛-1728= – ∛1728= -12

-12 is the cube root of (-1728).

(iii) (-2744)

Find the prime factors of the number 2744.

2744 = 14 × 14 × 14

Group the prime factors into each triplet.

2744 = (14 × 14 × 14)

Collect each one factor from each group.

14

Finally, find the product of each one factor from each group.

∛2744 = 14

∛-m = – ∛m

∛-2744 = – ∛2744= -14

-14 is the cube root of (-2744).

(iv) (-512)

Find the prime factors of the number 512.

512 = 8 × 8 × 8

Group the prime factors into each triplet.

512 = (8 × 8 × 8)

Collect each one factor from each group.

8

Finally, find the product of each one factor from each group.

∛512= 8

∛-m = – ∛m

∛-512= – ∛512= -8

-8 is the cube root of (-512).

4. Evaluate Cube Root of Product of Integers?

(i) ∛[27 × (-343)] (ii) ∛(64 × 216) (iii) ∛(125 × 216)

## Solution:

(i) ∛[27 × (-343)]

Firstly, apply the cube root to both integers.

∛ab = (∛a × ∛b)

∛[27 × (-343)] = ∛27 × ∛-343

Then, find the prime factors for each integer separately.

[∛{3 × 3 × 3}] × [∛{(-7) × (-7) × (-7)}]

Take each integer from the group in triplets and multiply them to get the cube root of a given number.

(3 × (-7)) = -21

-21 is the cube root of ∛[27 × (-343)].

(ii) ∛(64 × 216)

Firstly, apply the cube root to both integers.

∛ab = (∛a × ∛b)

∛(64 × 216) = ∛64 × ∛216

Then, find the prime factors for each integer separately.

[∛{4 × 4 × 4}] × [∛{6 × 6 × 6}]

Take each integer from the group in triplets and multiply them to get the cube root of a given number.

(4 × 6) = 24

24 is the cube root of ∛(64 × 216).

(iii) ∛(125 × 216)

Firstly, apply the cube root to both integers.

∛ab = (∛a × ∛b)

∛(125 × 216) = ∛125 × ∛216

Then, find the prime factors for each integer separately.

[∛{5 × 5 × 5}] × [∛{6 × 6 × 6}]

Take each integer from the group in triplets and multiply them to get the cube root of a given number.

(5 × 6) = 30

30 is the cube root of ∛(125 × 216).

5. Find the Cube Root of Decimal Number 4.096?

## Solution:

Convert the given decimal 4.096 into a fraction.

4.096 = 4096/1000

Now, apply the cube root to the fraction.

∛4096/1000

Apply the cube root to both integers.

∛(a/b) = (∛a)/(∛b)

∛4096/1000 = ∛4096/∛1000.

Then, find the prime factors for each integer separately.

∛(16 × 16 × 16)/∛(2 × 2 × 2 × 5 × 5 × 5)

Take each integer from the group in triplets to get the cube root of a given number.

(16)/(2 × 5) = 16/10

Convert the fraction into a decimal

16/10 = 1.6

1.6 is the cube root of 4.096.

6. Find the Cube Root of a Rational Number ∛(216/27)?

## Solution:

Firstly, apply the cube root to both integers.

∛(a/b) = (∛a)/(∛b)

∛(216/27) = ∛216/∛27

Then, find the prime factors for each integer separately.

[∛(6 × 6 × 6)]/[ ∛(3 × 3 × 3)]

Take each integer from the group in triplets to get the cube root of a given number.

6/3

6/3 is the cube root of ∛(216/2197).