Cube and Cube Roots

Q1: Multiple Choice Type:

i. The cube of 0.5 is:

Step 1: Cube means multiplying the number by itself three times. \[ (0.5)^3 = 0.5 \times 0.5 \times 0.5 \]Step 2: Multiply step-by-step: \[ 0.5 \times 0.5 = 0.25,\quad 0.25 \times 0.5 = 0.125 \]Answer: b. 0.125

ii. The cube of -4 is:

Step 1: Cube of -4 means: \[ (-4)^3 = (-4) \times (-4) \times (-4) \]Step 2: Multiply: \[ (-4) \times (-4) = 16,\quad 16 \times (-4) = -64 \]Answer: c. -64

iii. The smallest number by which 72 must be multiplied to obtain a perfect cube is:

Step 1: Prime factorise 72: \[ 72 = 2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2 \]Step 2: To make a cube, all prime powers should be multiples of 3. \[ 3^2 \Rightarrow \text{Needs one more 3 to become } 3^3 \]Step 3: Multiply by 3: \[ 72 \times 3 = 216,\quad 216 = 6^3 \]Answer: a. 3

iv. The smallest number by which 81 be divided to obtain a perfect cube is:

Step 1: Prime factorise 81: \[ 81 = 3 \times 3 \times 3 \times 3 = 3^4 \]Step 2: To make it a perfect cube, we need to make the power of 3 a multiple of 3. \[ 3^4 \div 3 = 3^3 = 27 \quad \text{(Perfect cube)} \]Answer: b. 3

Q2: Find the cube of:

i. 7

Step 1: \[ 7^3 = 7 \times 7 \times 7 = 343 \
Answer: 343

ii. 11

Step 1: \[ 11^3 = 11 \times 11 \times 11 = 1331 \]Answer: 1331

iii. 16

Step 1: \[ 16^3 = 16 \times 16 \times 16 = 4096 \]Answer: 4096

iv. 23

Step 1: \[ 23^3 = 23 \times 23 \times 23 = 12167 \]Answer: 12167

v. 31

Step 1: \[ 31^3 = 31 \times 31 \times 31 = 29791 \]Answer: 29791

vi. 42

Step 1: \[ 42^3 = 42 \times 42 \times 43 = 74088 \]Answer: 74088

vii. 54

Step 1: \[ 54^3 = 54 \times 54 \times 54 = 157464 \]Answer: 157464

Q3: Find which of the following are perfect cubes?

i. 243

Step 1: Prime factorisation of 243: \[ 243 = 3 \times 3 \times 3 \times 3 \times 3 = 3^5 \]Step 2: To be a perfect cube, all prime powers must be multiples of 3. \[ \text{But } 5 \text{ is not a multiple of 3} \]Answer: Not a perfect cube

ii. 588

Step 1: Prime factorisation of 588: \[ 588 = 2 \times 2 \times 3 \times 7 \times 7 = 2^2 \times 3 \times 7^2 \]Step 2: Powers are not multiples of 3.
Answer: Not a perfect cube

iii. 1331

Step 1: Try cube root: \[ \sqrt[3]{1331} = 11 \Rightarrow 11^3 = 1331 \]Step 2: Check by factorization: \[ 1331 = 11 \times 11 \times 11 = 11^3 \]Answer: Perfect cube

iv. 24000

Step 1: Prime factorisation: \[ 24000 = 2^7 \times 3 \times 5^3 \]Step 2: Powers of 2 and 3 are not multiples of 3: \[ \text{2^7 → Not divisible by 3, and 3^1 → Not cube} \]Answer: Not a perfect cube

v. 1728

Step 1: Prime factorisation: \[ 1728 = 2^6 \times 3^3 \]Step 2: All powers are divisible by 3
Step 3: Cube root: \[ \sqrt[3]{1728} = 12,\quad \text{since } 12^3 = 1728 \]Answer: Perfect cube

vi. 1938

Step 1: Prime factorisation: \[ 1938 = 2 \times 3 \times 17 \times 19 \]Step 2: None of the powers are cubes.
Answer: Not a perfect cube

Q4: Find the cubes of:

i. 2.1

Step 1: Multiply 2.1 three times \[ 2.1 \times 2.1 = 4.41,\quad 4.41 \times 2.1 = 9.261 \]Answer: 9.261

ii. 0.4

Step 1: Multiply 0.4 three times \[ 0.4 \times 0.4 = 0.16,\quad 0.16 \times 0.4 = 0.064 \]Answer: 0.064

iii. 1.6

Step 1: Multiply 1.6 three times \[ 1.6 \times 1.6 = 2.56,\quad 2.56 \times 1.6 = 4.096 \]Answer: 4.096

iv. 2.5

Step 1: Multiply 2.5 three times \[ 2.5 \times 2.5 = 6.25,\quad 6.25 \times 2.5 = 15.625 \]Answer: 15.625

v. 0.12

Step 1: Multiply 0.12 three times \[ 0.12 \times 0.12 = 0.0144,\quad 0.0144 \times 0.12 = 0.001728 \]Answer: 0.001728

vi. 0.02

Step 1: Multiply 0.02 three times \[ 0.02 \times 0.02 = 0.0004,\quad 0.0004 \times 0.02 = 0.000008 \]Answer: 0.000008

vii. 0.8

Step 1: Multiply 0.8 three times \[ 0.8 \times 0.8 = 0.64,\quad 0.64 \times 0.8 = 0.512 \]Answer: 0.512

Q5: Find the cubes of:

i. \(\frac{3}{7}\)

Step 1: Cube numerator and denominator separately: \[ \left(\frac{3}{7}\right)^3 = \frac{3^3}{7^3} = \frac{27}{343} \]Answer: \(\frac{27}{343}\)

ii. \(\frac{8}{9}\)

Step 1: Cube numerator and denominator: \[ \left(\frac{8}{9}\right)^3 = \frac{8^3}{9^3} = \frac{512}{729} \]Answer: \(\frac{512}{729}\)

iii. \(\frac{10}{13}\)

Step 1: Cube numerator and denominator: \[ \left(\frac{10}{13}\right)^3 = \frac{10^3}{13^3} = \frac{1000}{2197} \]Answer: \(\frac{1000}{2197}\)

iv. \(1\frac{2}{7}\)

Step 1: Convert to improper fraction: \[ 1\frac{2}{7} = \frac{9}{7} \]Step 2: Cube it: \[ \left(\frac{9}{7}\right)^3 = \frac{729}{343} = 2\frac{43}{343} \]Answer: \(2\frac{43}{343}\)

v. \(2\frac{1}{2}\)

Step 1: Convert to improper fraction: \[ 2\frac{1}{2} = \frac{5}{2} \]Step 2: Cube it: \[ \left(\frac{5}{2}\right)^3 = \frac{125}{8} = 15\frac{5}{8} \]Answer: \(15\frac{5}{8}\)

Q6: Find the cubes of:

i. -3

Step 1: Cube of -3 means multiplying it three times: \[ (-3)^3 = (-3) \times (-3) \times (-3) \]Step 2: First two negative numbers give positive: \[ (-3) \times (-3) = 9,\quad 9 \times (-3) = -27 \]Answer: -27

ii. -7

Step 1: Cube of -7: \[ (-7)^3 = (-7) \times (-7) \times (-7) = 49 \times (-7) = -343 \]Answer: -343

iii. -12

Step 1: Cube of -12: \[ (-12)^3 = (-12) \times (-12) \times (-12) = 144 \times (-12) = -1728 \]Answer: -1728

iv. -18

Step 1: Cube of -18: \[ (-18)^3 = (-18) \times (-18) \times (-18) = 324 \times (-18) = -5832 \]Answer: -5832

v. -25

Step 1: Cube of -25: \[ (-25)^3 = (-25) \times (-25) \times (-25) = 625 \times (-25) = -15625 \]Answer: -15625

vi. -30

Step 1: Cube of -30: \[ (-30)^3 = (-30) \times (-30) \times (-30) = 900 \times (-30) = -27000 \]Answer: -27000

vii. -50

Step 1: Cube of -50: \[ (-50)^3 = (-50) \times (-50) \times (-50) = 2500 \times (-50) = -125000 \]Answer: -125000

Q7: Which of the following are cubes of: i. an even number
ii. an odd number.
216, 729, 3375, 8000, 125, 343, 4096 and 9261

i. 216

Step 1: Find cube root: \[ \sqrt[3]{216} = \sqrt[3]{2 \times 2 \times 2 \times 3 \times 3 \times 3} = 2 \times 3 = 6 \]Step 2: 6 is an even number.
Answer: Even number

ii. 729

Step 1: Cube root: \[ \sqrt[3]{729} = \sqrt[3]{3 \times 3 \times 3 \times 3 \times 3 \times 3} = 3 \times 3 = 9 \]Step 2: 9 is an odd number.
Answer: Odd number

iii. 3375

Step 1: Cube root: \[ \sqrt[3]{3375} = \sqrt[3]{3 \times 3 \times 3 \times 5 \times 5 \times 5} = 3 \times 5 = 15 \]Step 2: 15 is an odd number.
Answer: Odd number

iv. 8000

Step 1: Cube root: \[ \sqrt[3]{8000} = \sqrt[3]{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 5 \times 5 \times 5} \\ = 2 \times 2 \times 5 = 20 \]Step 2: 20 is an even number.
Answer: Even number

v. 125

Step 1: Cube root: \[ \sqrt[3]{125} = \sqrt[3]{5 \times 5 \times 5} = 5 \]Step 2: 5 is an odd number.
Answer: Odd number

vi. 343

Step 1: Cube root: \[ \sqrt[3]{343} = = \sqrt[3]{7 \times 7 \times 7} = 7 \]Step 2: 7 is an odd number.
Answer: Odd number

vii. 4096

Step 1: Cube root: \[ \sqrt[3]{4096} = \sqrt[3]{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2} \\ = 2 \times 2 \times 2 \times 2 = 16 \]Step 2: 16 is an evan number.
Answer: Even number

viii. 9261

Step 1: Cube root: \[ \sqrt[3]{9261} = \sqrt[3]{3 \times 3 \times 3 \times 7 \times 7 \times 7} = 3 \times 7 = 21 \]Step 2: 21 is an odd number.
Answer: Odd number

Q8: Find the least number by which 1323 must be multiplied that the product a perfect cube.

Step 1: Find the prime factorisation of 1323 \[ 1323 \div 3 = 441 \quad \Rightarrow \quad 441 \div 3 = 147 \quad \Rightarrow \quad 147 \div 3 = 49 \quad \Rightarrow \quad 49 = 7 \times 7 \] So, \[ 1323 = 3 \times 3 \times 3 \times 7 \times 7 = 3^3 \times 7^2 \]Step 2: For a number to be a perfect cube, all prime powers must be multiples of 3
Here, \[ 3^3 \text{ is a cube already, but } 7^2 \text{ is not} \]Step 3: Multiply by one more 7 to make exponent of 7 equal to 3: \[ 1323 \times 7 = 9261 \]Now: \[ 9261 = 3^3 \times 7^3 = (3 \times 7)^3 = 21^3 \]Answer: 7

The least number is 7 to make 1323 a perfect cube.

Q9: Find the smallest number by which 8768 must be multiplied that the product a perfect cube.

Step 1: Perform prime factorisation of 8768
We start by dividing by 2: \[ 8768 \div 2 = 4384 \quad \div 2 = 2192 \quad \div 2 = 1096 \quad \div 2 = 548 \quad \div 2 = 274 \quad \div 2 = 137 \] Now 137 is a prime number.
So, prime factors of 8768 are: \[ 8768 = 2^6 \times 137 \]Step 2: Group the powers to check for cube formation
We have:
– \(2^6\) → cube-ready (since \(6\) is a multiple of 3)
– \(137^1\) → not cube-ready
Step 3: 137 is the least number by which 8768 must be multiplied that the product a perfect cube.
Answer: 137

Q10: Find the smallest number by which 27783 should be multiplied to get a perfect cube number.

Step 1: Find the prime factorisation of 27783
Check divisibility by 3: \[ 27783 \div 3 = 9261 \quad \div 3 = 3087 \quad \div 3 = 1029 \quad \div 3 = 343 \] Now, \[ 343 = 7 \times 7 \times 7 = 7^3 \]So, we have: \[ 27783 = 3^4 \times 7^3 \]Step 2: To make a perfect cube, all powers must be multiples of 3
– \(7^3\) is already a perfect cube
– \(3^4\) → needs to become \(3^6\) (next multiple of 3 after 4)
→ So we must multiply by \(3^2 = 9\)
Step 3: Multiply by 9: \[ 27783 \times 9 = 250047 \] Check: \[ 250047 = (3^6) \times (7^3) = (3^2 \times 7)^3 = (63)^3 \]Answer: 9

The smallest number is 9 to make 27783 a perfect cube.

Q11: With what least number should 8640 be divided so that the quotient is a perfect cube?

Step 1: Perform prime factorisation of 8640 \[ 8640 = 2^6 \times 3^3 \times 5 \]Step 2: For a number to be a perfect cube, all prime exponents must be multiples of 3
We analyse each prime power:
– \(2^6\) ✅ (6 is a multiple of 3)
– \(3^3\) ✅ (3 is a multiple of 3)
– \(5^1\) ❌ (1 is not a multiple of 3)
Step 3: To make it a perfect cube, we must remove \(5^1\) (as 1 is not divisible by 3)
So, divide 8640 by 5: \[ 8640 \div 5 = 1728 \]Now, \[ 1728 = 2^6 \times 3^3 = (2^2 \times 3)^3 = (12)^3 \]Step 4: The quotient is a perfect cube.
Answer: 5

The least number to divide 8640 by to get a perfect cube is 5.

Q12: Which is the smallest number that should be multiplied to 77175 to make it a perfect cube?

Step 1: Find the prime factorisation of 77175
Divide by 5: \[ 77175 \div 5 = 15435 \quad \div 5 = 3087 \]Now divide 3087 by 3 repeatedly: \[ 3087 \div 3 = 1029 \quad \div 3 = 343 \]Now factor 343: \[ 343 = 7 \times 7 \times 7 = 7^3 \]So the full factorisation: \[ 77175 = 3^2 \times 5^2 \times 7^3 \]Step 2: For a number to be a perfect cube, each exponent must be a multiple of 3
We need to adjust:
– \(3^2 \rightarrow\) multiply by one more 3 to make \(3^3\)
– \(5^2 \rightarrow\) multiply by one more 5 to make \(5^3\)
– \(7^3\) is already a perfect cube ✅
Step 3: Multiply by \(3 \times 5 = 15\) \[ 77175 \times 15 = 1157625 \]Now: \[ 1157625 = (3 \times 5 \times 7)^3 = (105)^3 \]Answer: 15

The smallest number to multiply 77175 and make it a perfect cube is 15.