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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).