Cubes and Cube Roots: Important Questions and Answers for Class 8

Cubes and Cube Root Class 8 Important Questions are available in this Maths article. Cubes and Cube Root Class 8 Important Questions are very useful to solve the problems easily. This article helps the students to know the key questions and answers about Cubes and Cube Roots. Cubes and cube roots show numbers multiplied three times and their reverse. Our subject experts have provided detailed solutions for these problems based on the CBSE syllabus and the NCERT textbook. This material helps students revise the chapter easily and perform well in the final examination.

Table of Contents

• Exercise 7.1: Introduction to Cubes and Perfect Cubes
• Exercise 7.2: Properties of Cubes and Cube Numbers
• Exercise 7.3: Finding Cube Roots
• Exercise 7.4: Finding Cube Roots Using Prime Factorization
• Mixed Exercise Questions
• Tips for Understanding Cubes and Cube Roots
• Most Repeated Board Questions on Cubes and Cube Roots

Exercise 7.1: Introduction to Cubes and Perfect Cubes

This exercise helps students understand what cubes are and recognize perfect cube numbers.

Question 1: What is a Cube? 

Answer: A cube is when you multiply a number by itself three times. We write this using a small 3 as a superscript.

If the number is n, then n cubed is written as n³ = n × n × n

Examples:

• 2 cubed = 2³ = 2 × 2 × 2 = 8
• 3 cubed = 3³ = 3 × 3 × 3 = 27
• 4 cubed = 4³ = 4 × 4 × 4 = 64
• 5 cubed = 5³ = 5 × 5 × 5 = 125

If you have a cube-shaped box with sides of 2 cm each, the volume (space inside) is 2³ = 8 cubic cm.

Read more: Important Questions on Square and Square Roots - Class 8

Question 2: What is a Perfect Cube? Give Examples

Answer: A perfect cube is a number that is the cube of a whole number. When you cube a whole number, you get a perfect cube.

Perfect cubes from 1 to 1000:

• 1³ = 1
• 2³ = 8
• 3³ = 27
• 4³ = 64
• 5³ = 125
• 6³ = 216
• 7³ = 343
• 8³ = 512
• 9³ = 729
• 10³ = 1000

So the perfect cubes from 1 to 1000 are: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000

Question 3: Is 50 a Perfect Cube?

Answer: No, 50 is not a perfect cube.

Let's find which numbers when cubed give values close to 50.

3³ = 27 (too small) 4³ = 64 (too big)

Since 50 is between 27 and 64, and there's no whole number that when cubed gives 50, it is not a perfect cube.

Question 4: Find the Cube of 7

Solution: 7³ = 7 × 7 × 7

First: 7 × 7 = 49 Then: 49 × 7 = 343

Answer: 7³ = 343

Question 5: Calculate 15³

Solution: 15³ = 15 × 15 × 15

First: 15 × 15 = 225 Then: 225 × 15 = ?

Let me calculate: 225 × 15 = 225 × 10 + 225 × 5 = 2250 + 1125 = 3375

Answer: 15³ = 3375

Question 6: List All Perfect Cubes Between 1 and 100

Answer: Finding perfect cubes between 1 and 100:

1³ = 1

2³ = 8

3³ = 27

4³ = 64

Perfect cubes between 1 and 100: 1, 8, 27, 64

Exercise 7.2: Properties of Cubes and Cube Numbers

Question 7: What Pattern Do You Notice in Cubes of Numbers Ending in 0, 1, 2, Etc.?

Answer: There's a clear pattern: the cube of a number ends with the same digit as the number itself.

Examples:

• Numbers ending in 0: 10³ = 1000 (ends in 0), 20³ = 8000 (ends in 0)
• Numbers ending in 1: 11³ = 1331 (ends in 1), 21³ = 9261 (ends in 1)
• Numbers ending in 2: 2³ = 8, 12³ = 1728 (ends in 8 - this is different!)
• Numbers ending in 3: 3³ = 27 (ends in 7 - this is different!)
• Numbers ending in 4: 4³ = 64 (ends in 4), 14³ = 2744 (ends in 4)
• Numbers ending in 5: 5³ = 125 (ends in 5), 15³ = 3375 (ends in 5)
• Numbers ending in 6: 6³ = 216 (ends in 6), 16³ = 4096 (ends in 6)
• Numbers ending in 7: 7³ = 343 (ends in 3 - this is different!)
• Numbers ending in 8: 8³ = 512 (ends in 2 - this is different!)
• Numbers ending in 9: 9³ = 729 (ends in 9), 19³ = 6859 (ends in 9)

Pattern: Some digits (0, 1, 4, 5, 6, 9) stay the same, while others change (2→8, 3→7, 7→3, 8→2)

Question 8: Cube of an Even Number is Always Even. True or False?

Answer: True. The cube of an even number is always even.

An even number can be written as 2n. When we cube it: (2n)³ = 2n × 2n × 2n = 8n³ = 2(4n³)

Examples:

• 2³ = 8 (even)
• 4³ = 64 (even)
• 6³ = 216 (even)
• 8³ = 512 (even)

Question 9: Cube of an Odd Number is Always Odd. True or False?

Answer: True. The cube of an odd number is always odd.

An odd number can be written as (2n + 1). When we cube it, we get an odd number.

Examples:

• 1³ = 1 (odd)
• 3³ = 27 (odd)
• 5³ = 125 (odd)
• 7³ = 343 (odd)
• 9³ = 729 (odd)

Question 10: Find the Cube of Negative Numbers: (-2)³ and (-5)³

Solution: When we cube negative numbers, the answer is also negative.

(-2)³ = (-2) × (-2) × (-2) = 4 × (-2) = -8

(-5)³ = (-5) × (-5) × (-5) = 25 × (-5) = -125

Pattern: (−n)³ = −(n³)

Examples:

• (-1)³ = -1
• (-3)³ = -27
• (-4)³ = -64
• (-10)³ = -1000

Question 11: What is the Cube of a Fraction? Find (2/3)³

Solution: To cube a fraction, we cube both the numerator and denominator.

(2/3)³ = (2/3) × (2/3) × (2/3) = (2×2×2)/(3×3×3) = 8/27

Answer: (2/3)³ = 8/27

Question 12: Calculate (-1/2)³

Solution: (-1/2)³ = (-1/2) × (-1/2) × (-1/2)

= ((-1) × (-1) × (-1))/(2 × 2 × 2) = -1/8

Answer: (-1/2)³ = -1/8

Exercise 7.3: Finding Cube Roots

Question 13: What is a Cube Root?

Answer: A cube root is the number that when multiplied by itself three times gives you the original number.

If a³ = b, then the cube root of b is a. We write this as ∛b = a

If you know the volume of a cube-shaped box is 125 cubic units, the cube root of 125 tells you that each side of the box is 5 units.

Examples:

• Cube root of 8 = 2 (because 2³ = 8) → ∛8 = 2
• Cube root of 27 = 3 (because 3³ = 27) → ∛27 = 3
• Cube root of 64 = 4 (because 4³ = 64) → ∛64 = 4
• Cube root of 125 = 5 (because 5³ = 125) → ∛125 = 5

Question 14: Find the Cube Root of 216

Answer: We need to find a number that when cubed equals 216.

Which number cubed gives 216?

6³ = 6 × 6 × 6 = 216

Answer: ∛216 = 6

Question 15: Find ∛343

Answer: We need to find which number when cubed gives 343.

Let's check: 7³ = 7 × 7 × 7 = 343

Answer: ∛343 = 7

Question 16: Find the Cube Root of 1000

Answer: We need a number that when cubed equals 1000.

10³ = 10 × 10 × 10 = 1000

Answer: ∛1000 = 10

Question 17: Find ∛(-27)

We need a number that when cubed equals -27.

(-3)³ = (-3) × (-3) × (-3) = -27

Answer: ∛(-27) = -3

The cube root of a negative number is negative.

Question 18: Find ∛(8/125)

We need a number that when cubed equals 8/125.

What cubed gives 8/125?

(2/5)³ = (2/5) × (2/5) × (2/5) = (2×2×2)/(5×5×5) = 8/125

Answer: ∛(8/125) = 2/5

Exercise 7.4: Finding Cube Roots Using Prime Factorization

Question 19: What is Prime Factorization?

Answer: Prime factorization means breaking down a number into prime numbers that multiply to give the original number.

Prime numbers are numbers that can only be divided by 1 and themselves.

Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, ...

Example: Prime factorization of 24: 24 = 2 × 12 = 2 × 2 × 6 = 2 × 2 × 2 × 3 = 2³ × 3

Example: Prime factorization of 72: 72 = 8 × 9 = 2³ × 3²

Question 20: Find ∛8 Using Prime Factorization

Solution: Step 1: Find prime factorization of 8 8 = 2 × 4 = 2 × 2 × 2 = 2³

Step 2: To find cube root, group factors in threes 8 = 2³

Step 3: Take one factor from each group ∛8 = 2

Answer: ∛8 = 2

Question 21: Find ∛343 Using Prime Factorization

Solution: Step 1: Find prime factorization of 343 343 = 7 × 49 = 7 × 7 × 7 = 7³

Step 2: Group factors in threes 343 = 7³

Step 3: Take one from each group ∛343 = 7

Answer: ∛343 = 7

Question 22: Find ∛216 Using Prime Factorization

Solution: Step 1: Find prime factorization of 216 216 = 2 × 108 = 2 × 2 × 54 = 2 × 2 × 2 × 27 = 2³ × 27 = 2³ × 3³

Step 2: Group factors in threes 216 = 2³ × 3³ = (2 × 3)³ = 6³

Step 3: Take one from each group ∛216 = 2 × 3 = 6

Answer: ∛216 = 6

Question 23: Find ∛1728 Using Prime Factorization

Solution: Step 1: Find prime factorization of 1728 1728 = 2 × 864 = 2 × 2 × 432 = 2 × 2 × 2 × 216 = 2³ × 216 = 2³ × 2³ × 27 = 2³ × 2³ × 3³

1728 = 12 × 144 = 12 × 12 × 12 = 12³

Or: 1728 = 2³ × 2³ × 3³ = (2 × 2 × 3)³ = (4 × 3)³ = 12³

Step 1: We have groups of three 1728 = (2 × 2 × 3)³

Step 2: Take one from each group ∛1728 = 2 × 2 × 3 = 12

Answer: ∛1728 = 12

Question 24: Check if 64 is a Perfect Cube and Find Its Cube Root

Solution: Step 1: Find prime factorization of 64 64 = 2 × 32 = 2 × 2 × 16 = 2 × 2 × 2 × 8 = 2 × 2 × 2 × 2 × 4 = 2 × 2 × 2 × 2 × 2 × 2 = 2⁶

Step 2: Check if exponents are divisible by 3 Exponent is 6, and 6 ÷ 3 = 2

So 64 is a perfect cube.

Step 3: Find cube root 64 = 2⁶ = (2²)³ = 4³

∛64 = 4

Answer: 64 is a perfect cube, and ∛64 = 4

Question 25: Find ∛(-1000)

Solution: Step 1: Find prime factorization of 1000 1000 = 10 × 100 = 10 × 10 × 10 = 10³

Step 2: For negative number (-1000) = -(1000) = -(10³) = (-10)³

Step 3: Cube root ∛(-1000) = -10

Answer: ∛(-1000) = -10

Mixed Exercise Questions

Question 26: Is 128 a Perfect Cube? Why or Why Not?

Answer: Let's find the prime factorization of 128.

128 = 2 × 64 = 2 × 2 × 32 = 2 × 2 × 2 × 16 = 2 × 2 × 2 × 2 × 8 = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2⁷

For a number to be a perfect cube, all prime factors must have exponents divisible by 3.

Here, the exponent of 2 is 7. 7 ÷ 3 = 2 remainder 1 (not divisible by 3)

Answer: No, 128 is not a perfect cube because the exponent of 2 is 7, which is not divisible by 3.

Question 27: What Number Should Multiply 24 to Make It a Perfect Cube?

Solution: Step 1: Find prime factorization of 24 24 = 2³ × 3¹

Step 2: For a perfect cube, all exponents must be divisible by 3

• Exponent of 2 is 3 (already divisible by 3) 
• Exponent of 3 is 1 (need to make it 3)

Step 3: We need 2 more factors of 3 So we should multiply by 3² = 9

Step 4: Check 24 × 9 = 216 = 2³ × 3³ = 6³ 

Answer: We should multiply 24 by 9 to get 216, which is 6³

Question 28: Find ∛729

Solution: Let's find prime factorization:

729 = 3 × 243 = 3 × 3 × 81 = 3 × 3 × 3 × 27 = 3 × 3 × 3 × 3 × 9 = 3 × 3 × 3 × 3 × 3 × 3 = 3⁶ = (3²)³ = 9³

∛729 = 9

Answer: ∛729 = 9

Question 29: Compare: 2³ + 3³ vs (2+3)³

Solution: Left side: 2³ + 3³ = 8 + 27 = 35

Right side: (2+3)³ = 5³ = 125

35 ≠ 125

Answer: 2³ + 3³ = 35, but (2+3)³ = 125. They are not equal.

Cube of a sum is NOT equal to sum of cubes.

Question 30: Simplify: ∛(1000/125)

Solution: Method 1: Simplify fraction first 1000/125 = 8

∛8 = 2

Method 2: Apply cube root to numerator and denominator ∛(1000/125) = ∛1000 / ∛125 = 10 / 5 = 2

Answer: ∛(1000/125) = 2

Tips for Understanding Cubes and Cube Roots

Tip 1: Memorize Cubes of Numbers 1 to 10

Learning these by heart makes solving problems much faster.

• 1³ = 1
• 2³ = 8
• 3³ = 27
• 4³ = 64
• 5³ = 125
• 6³ = 216
• 7³ = 343
• 8³ = 512
• 9³ = 729
• 10³ = 1000

Once you know these, finding cube roots of these numbers becomes instant. For example, if you see 343, you immediately know it's 7³, so ∛343 = 7.

Most Repeated Board Questions on Cubes and Cube Roots

Question 1: Finding Cubes of Integers

Find the cube of 12.

Solution: 12³ = 12 × 12 × 12 = 144 × 12 = 1728

Question 2: Identifying Perfect Cubes

Solution: 512 = 2⁹ = (2³)³ = 8³ So yes, it's a perfect cube, and ∛512 = 8

Question 3: Using Prime Factorization to Find Cube Roots

Find ∛1728 using prime factorization method"

Solution : 1728 = 2⁶ × 3³ = (2² × 3)³ = 12³ ∛1728 = 12

Question 4: Cubes and Cube Roots of Negative Numbers

Find (-6)³ and ∛(-216)

Solution : (-6)³ = -216 ∛(-216) = -6

Question 5: Finding What to Multiply to Make Perfect Cube

What number should 32 be multiplied by to make it a perfect cube?

Solution : 32 = 2⁵ Need: 2⁶ (so exponent is divisible by 3) Multiply by 2¹ = 2 32 × 2 = 64 = 4³

Question 6: Cube Roots of Fractions

Find ∛(64/343)

Solution : ∛(64/343) = ∛64 / ∛343 = 4 / 7

Question 7: Volume Problems with Cubes

A cubic container has volume 1000 cm³. Find its side length

Solution : Side length = ∛1000 = 10 cm

Question 8: Comparing Cubes and Cube Operations

Compare: (2 + 3)³ and 2³ + 3³

Solution : (2+3)³ = 5³ = 125 2³ + 3³ = 8 + 27 = 35 They are different, showing (a+b)³ ≠ a³ + b³

Tests understanding of operations and properties

Question 9: Perfect Cube Verification

Verify that 729 is a perfect cube

Solution : 729 = 3⁶ = (3²)³ = 9³ Yes, it's a perfect cube

Question 10: Simplifying Cube Root Expressions

Simplify: ∛(8 × 27)

Solution : ∛(8 × 27) = ∛8 × ∛27 = 2 × 3 = 6

Or: 8 × 27 = 216 = 6³, so ∛216 = 6

Most Common Examination Questions (Board Exams)

1 - 2 Mark Questions (Very Frequently Asked):

1. Find the cube of 5 Answer: 125
2. Is 100 a perfect cube? Answer: No, because ∛100 ≈ 4.64 (not a whole number)
3. Find ∛(−8) Answer: −2
4. What is 10³? Answer: 1000
5. Write the cube of 3 Answer: 27
6. Find the cube root of 64 Answer: 4
7. Is 216 a perfect cube? Answer: Yes, 216 = 6³
8. Find (−3)³ Answer: −27
9. What is the cube root of 1? Answer: 1
10. Express 125 as a perfect cube Answer: 125 = 5³

3 - 4 Mark Questions (Frequently Asked):

1. Find the cube of 12 Solution: 12³ = 12 × 12 × 12 = 1728
2. Check whether 256 is a perfect cube Solution: 256 = 2⁸ Exponent 8 is not divisible by 3 Not a perfect cube
3. Find ∛512 using prime factorization Solution: 512 = 2⁹ = (2³)³ = 8³ ∛512 = 8
4. Find ∛1331 Solution: 1331 = 11³ ∛1331 = 11
5. What should 54 be multiplied by to make a perfect cube? Solution: 54 = 2 × 3³ Need one more factor of 2 54 × 2 = 108 = 2² × 3³ But wait, 2² needs another 2 54 × 4 = 216 = 2³ × 3³ = 6³ Answer: Multiply by 4
6. Find ∛(27/64) Solution: ∛(27/64) = ∛27 / ∛64 = 3 / 4
7. Show that 10³ + 20³ ≠ (10 + 20)³ Solution: 10³ + 20³ = 1000 + 8000 = 9000 (10 + 20)³ = 30³ = 27000 9000 ≠ 27000
8. Find the side of a cube if its volume is 1728 cm³ Solution: Side = ∛1728 = 12 cm
9. Find ∛(−343) Solution: (−7)³ = −343 So ∛(−343) = −7
10. Is 128 a perfect cube? Give reason Solution: 128 = 2⁷ Exponent 7 is not divisible by 3 Not a perfect cube

5 - 6 Mark Questions (Less Frequent but Important):

1. Find which perfect cube numbers lie between 1 and 1000, then find their cube roots Solution: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000 Cube roots: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
2. Using prime factorization, determine if 3375 is a perfect cube. If yes, find its cube root Solution: 3375 = 3³ × 5³ = (3 × 5)³ = 15³ Yes, it's a perfect cube with cube root 15
3. A cubic box has a volume of 8000 cubic units. Find the length of its side. Also find the area of one face Solution: Side = ∛8000 = 20 units Area of one face = 20² = 400 square units
4. Compare the cubes of successive numbers. What pattern do you notice? Solution: Differences between consecutive cubes: 2³ − 1³ = 8 − 1 = 7 3³ − 2³ = 27 − 8 = 19 4³ − 3³ = 64 − 27 = 37 Pattern: Differences increase (odd number pattern: 7, 19, 37...)
5. Find x if x³ = 2197 Solution: x = ∛2197 2197 = 13³ So x = 13
6. Simplify: ∛(216) + ∛(343) − ∛(125) Solution: = 6 + 7 − 5 = 8
7. Find the smallest number by which 648 should be divided to make it a perfect cube Solution: 648 = 2³ × 3⁴ Need exponents divisible by 3 Divide by 3¹ = 3 648 ÷ 3 = 216 = 2³ × 3³ = 6³
8. Prove that (a + b)³ ≠ a³ + b³ using an example Solution: Let a = 2, b = 3 (2 + 3)³ = 5³ = 125 2³ + 3³ = 8 + 27 = 35 125 ≠ 35 ✓
9. A number when increased by 1 becomes a perfect cube. If the number is 26, find the cube root of the new number Solution: New number = 26 + 1 = 27 = 3³ Cube root = 3
10. Create a table of cubes from 1 to 15 and identify patterns in the last digit Solution: Show 1³=1, 2³=8, 3³=27, ... 15³=3375 Last digits: 1, 8, 7, 4, 5, 6, 3, 2, 9, 0, 1, 8, 7, 4, 5 Pattern repeats every 10 numbers