Cubes And Cube Roots — for Class 8

What Are Cubes and Cube Roots?

You already know squares and square roots from Class 7. Cubes are the natural next step — instead of multiplying a number by itself twice, we multiply it three times. So when we say $5^3$ , we mean $5 \times 5 \times 5 = 125$ .

The word “cube” comes from geometry. If you take a cube (the 3D shape) with side length 5 cm, its volume is exactly 125 cm³. That’s not a coincidence — that’s where the name comes from.

Cube roots reverse this process. The cube root of 125 is 5, written as $\sqrt[3]{125} = 5$ , because $5^3 = 125$ . We’re asking: “what number, multiplied by itself three times, gives us this result?”

This chapter has consistent weightage in CBSE Class 8 board exams and appears regularly in olympiad prelims. Master it here and the problems become almost mechanical.

Key Terms and Definitions

Perfect Cube: A number that can be expressed as $n^3$ where $n$ is a natural number. For example, 8 is a perfect cube because $8 = 2^3$ . But 9 is not a perfect cube — no natural number cubed equals 9.

Cube of a Number: The result of multiplying a number by itself three times. Written as $n^3$ . So $4^3 = 4 \times 4 \times 4 = 64$ .

Cube Root: The inverse operation of cubing. If $n^3 = m$ , then $\sqrt[3]{m} = n$ . Symbol is $\sqrt[3]{\phantom{x}}$ (radical with a small 3).

Prime Factorisation Method: The standard NCERT method to find cube roots — break the number into prime factors, then group them in sets of three.

First 10 Perfect Cubes — Memorise These

$n$	$n^3$
1	1
2	8
3	27
4	64
5	125
6	216
7	343
8	512
9	729
10	1000

Notice the unit digits of cubes: 1→1, 2→8, 3→7, 4→4, 5→5, 6→6, 7→3, 8→2, 9→9, 0→0. The unit digit of the cube tells you the unit digit of the cube root. This trick saves time in MCQs.

Core Concepts

Properties of Cubes

Understanding these properties helps you solve problems faster than just brute-force calculation.

Property 1 — Sign preservation: The cube of a positive number is positive; the cube of a negative number is negative.

(-3)^3 = -27 \quad \text{and} \quad (3)^3 = 27

This is different from squares, where negatives become positive. Cubes keep their sign.

Property 2 — Odd cubes are odd, even cubes are even: $3^3 = 27$ (odd), $4^3 = 64$ (even). The parity is preserved.

Property 3 — Cubes of fractions:

\left(\frac{a}{b}\right)^3 = \frac{a^3}{b^3}

So $\left(\frac{2}{3}\right)^3 = \frac{8}{27}$ .

The Sum of Consecutive Odd Numbers Pattern

There’s a beautiful pattern NCERT highlights:

1^3 = 1

2^3 = 3 + 5 = 8

3^3 = 7 + 9 + 11 = 27

4^3 = 13 + 15 + 17 + 19 = 64

The cube of $n$ equals the sum of $n$ consecutive odd numbers starting from $n^2 - n + 1$ . You won’t need to derive this in CBSE, but if you see a pattern-based question, this is the key.

Finding Cube Roots by Prime Factorisation

This is the NCERT standard method and the one you must show in step-marked questions.

The Rule: Express the number as a product of primes. If every prime factor appears exactly 3 times (or in multiples of 3), the number is a perfect cube. Group the factors in triplets, take one factor from each group — that gives you the cube root.

\sqrt[3]{n} = \sqrt[3]{p^3 \times q^3 \times r^3 \times \ldots} = p \times q \times r \times \ldots

Solved Examples

Example 1 — Easy (CBSE Level): Is 216 a perfect cube?

Step 1: Prime factorise 216.

216 = 2 \times 108 = 2 \times 2 \times 54 = 2 \times 2 \times 2 \times 27 = 2 \times 2 \times 2 \times 3 \times 3 \times 3

216 = 2^3 \times 3^3

Step 2: Check if all primes appear in triplets. Both 2 and 3 appear exactly 3 times. ✓

Step 3: Find the cube root.

\sqrt[3]{216} = 2 \times 3 = 6

Answer: Yes, 216 is a perfect cube, and $\sqrt[3]{216} = 6$ .

Example 2 — Easy (CBSE Level): Find the smallest number by which 392 must be multiplied to make it a perfect cube.

Why this type of question? CBSE loves “smallest multiplier/divisor” problems. The logic is: after prime factorisation, we need every prime to appear in a multiple of 3. Whichever prime is “incomplete” in its triplet — that’s what we need to multiply by.

Step 1: Factorise 392.

392 = 2 \times 196 = 2 \times 2 \times 98 = 2 \times 2 \times 2 \times 49 = 2^3 \times 7^2

Step 2: Check triplets. The factor 2 forms a complete triplet ( $2^3$ ). But 7 appears only twice ( $7^2$ ). We need one more 7 to complete the triplet.

Step 3: Multiply by 7.

392 \times 7 = 2744 = 2^3 \times 7^3

\sqrt[3]{2744} = 2 \times 7 = 14

Answer: Multiply by 7.

Example 3 — Medium (CBSE Level): Find the smallest number by which 2916 must be divided to make it a perfect cube.

Step 1: Factorise 2916.

2916 = 2^2 \times 3^6 = 4 \times 729

Let’s verify: $2916 \div 2 = 1458$ , $\div 2 = 729$ , $\div 3 = 243$ , $\div 3 = 81$ , $\div 3 = 27$ , $\div 3 = 9$ , $\div 3 = 3$ , $\div 3 = 1$ .

So $2916 = 2^2 \times 3^6$ .

Step 2: $3^6 = (3^3)^2$ — perfect triplets. ✓ But $2^2$ is incomplete — 2 appears only twice, needs one more 2 to complete the triplet.

Step 3: If we can’t add factors, we divide. Dividing by $2^2 = 4$ removes the incomplete group entirely.

2916 \div 4 = 729 = 3^6 = (3^2)^3

\sqrt[3]{729} = 3^2 = 9

Answer: Divide by 4. The resulting perfect cube is 729, with cube root 9.

Example 4 — Medium: Find the cube root of $-13824$ .

When the number is negative, ignore the sign first, find the cube root of the positive number, then attach the negative sign.

13824 = 2^3 \times 2^3 \times 3^3 \times \ldots

Let’s factorise: $13824 \div 2 = 6912 \div 2 = 3456 \div 2 = 1728 \div 2 = 864 \div 2 = 432 \div 2 = 216 \div 2 = 108 \div 2 = 54 \div 2 = 27$ .

So $13824 = 2^9 \times 3^3 = (2^3)^3 \times 3^3$ .

\sqrt[3]{13824} = 2^3 \times 3 = 8 \times 3 = 24

Therefore: $\sqrt[3]{-13824} = -24$

Example 5 — Hard: Find the cube root of $\frac{-512}{2197}$ .

For fractions, apply cube root separately to numerator and denominator.

\sqrt[3]{\frac{-512}{2197}} = \frac{\sqrt[3]{-512}}{\sqrt[3]{2197}}

$512 = 2^9 = (2^3)^3$ , so $\sqrt[3]{512} = 8$ , giving $\sqrt[3]{-512} = -8$ .

$2197 = 13^3$ , so $\sqrt[3]{2197} = 13$ .

\sqrt[3]{\frac{-512}{2197}} = \frac{-8}{13}

Exam-Specific Tips

CBSE Class 8 Pattern: This chapter typically contributes 3-5 marks in periodic tests. Expect one “is it a perfect cube?” question, one “smallest number to multiply/divide” question, and possibly one cube root by prime factorisation. Show all factorisation steps — CBSE awards 1 mark per step in structured solutions.

For Board Exams: Always show the prime factorisation tree or the division method explicitly. Writing just “216 = $2^3 \times 3^3$ ” without showing the division steps can cost you a step mark.

Olympiad/NTSE Tip: The unit digit trick (see Key Terms section above) lets you eliminate 3 out of 4 options in MCQs instantly. For example, if the question asks for $\sqrt[3]{17576}$ and the options are 25, 26, 27, 28 — the number ends in 6, so the cube root must end in 6. Only 26 ends in 6. Done.

Shortcut for Two-Digit Cube Roots: For perfect cubes between 1000 and 1,000,000, you can estimate using the unit digit trick plus the tens digit from the cube table. This is useful for mental maths but stick to prime factorisation in written exams.

Common Mistakes to Avoid

Mistake 1 — Confusing square root with cube root: $\sqrt{64} = 8$ but $\sqrt[3]{64} = 4$ . These are completely different operations. In exams, read the radical symbol carefully — the small 3 inside makes all the difference.

Mistake 2 — Wrong grouping in prime factorisation: Students sometimes group factors in pairs (like for square roots) instead of triplets. For cube roots, you MUST group in sets of 3. If you have $2^6$ , that’s two complete triplets ( $2^3 \times 2^3$ ), contributing $2 \times 2 = 4$ to the cube root.

Mistake 3 — Forgetting the negative sign: $\sqrt[3]{-27} = -3$ , not 3. Unlike square roots (which have no real solution for negatives), cube roots of negative numbers are simply negative. Don’t leave the negative sign behind.

Mistake 4 — Wrong answer in “smallest divisor” problems: When a prime appears twice (like $p^2$ ), students often say “divide by $p$ ” to leave $p^1$ , then add $p^2$ to get $p^3$ . No — if you’re dividing, you need to remove the incomplete group entirely. Divide by $p^2$ , not $p$ .

Mistake 5 — Assuming any odd number is a perfect cube: 27 is a perfect cube, 9 is not. Being odd has nothing to do with it. Always verify through prime factorisation.

Practice Questions

Q1. Check whether 1728 is a perfect cube. If yes, find its cube root.

$1728 = 2^6 \times 3^3 = (2^2)^3 \times 3^3$ . All primes in triplets. $\sqrt[3]{1728} = 2^2 \times 3 = 4 \times 3 = \mathbf{12}$ .

Q2. Find the smallest number by which 108 must be multiplied to get a perfect cube.

$108 = 2^2 \times 3^3$ . The factor 2 appears only twice — need one more 2. Multiply by 2. Then $216 = 2^3 \times 3^3$ , $\sqrt[3]{216} = 6$ .

Q3. Find the smallest number by which 1536 must be divided to make it a perfect cube.

$1536 = 2^9 \times 3$ . The factor 3 appears only once — removing it makes the number a perfect cube. Divide by 3. Then $512 = 2^9 = (2^3)^3$ , $\sqrt[3]{512} = 8$ .

Q4. Evaluate: $\sqrt[3]{-1000}$

$1000 = 10^3$ , so $\sqrt[3]{1000} = 10$ . Therefore $\sqrt[3]{-1000} = \mathbf{-10}$ .

Q5. Find the cube root of $\frac{64}{125}$ .

\sqrt[3]{\frac{64}{125}} = \frac{\sqrt[3]{64}}{\sqrt[3]{125}} = \frac{4}{5}

Q6. Three numbers are in ratio 1 : 2 : 3. If their product is 1296, find the numbers.

Let the numbers be $k$ , $2k$ , $3k$ . Then $k \times 2k \times 3k = 6k^3 = 1296$ . So $k^3 = 216$ , giving $k = 6$ . The numbers are 6, 12, 18.

Q7. Is 9000 a perfect cube? If not, find the smallest number to multiply to make it a perfect cube.

$9000 = 2^3 \times 3^2 \times 5^3$ . The factor 3 appears only twice. Multiply by 3 to get $3^3$ . Then $27000 = 2^3 \times 3^3 \times 5^3 = 30^3$ , $\sqrt[3]{27000} = 30$ .

Q8. Find the value of $\sqrt[3]{0.008}$ .

Write $0.008 = \frac{8}{1000}$ . Then $\sqrt[3]{\frac{8}{1000}} = \frac{\sqrt[3]{8}}{\sqrt[3]{1000}} = \frac{2}{10} = \mathbf{0.2}$ .

Q9. If the volume of a cube is $15625 \text{ cm}^3$ , find the length of its side.

Side = $\sqrt[3]{15625}$ . Factorise: $15625 = 5^6 = (5^2)^3$ . So side = $5^2 = \mathbf{25 \text{ cm}}$ .

Q10. Find the smallest perfect cube greater than 500.

$7^3 = 343$ (less than 500), $8^3 = 512$ (greater than 500). The answer is 512.

Frequently Asked Questions

What is the cube root of 1? $\sqrt[3]{1} = 1$ , because $1 \times 1 \times 1 = 1$ . Also, $\sqrt[3]{0} = 0$ .

Can negative numbers have cube roots? Yes — this is a key difference from square roots. $\sqrt[3]{-8} = -2$ because $(-2)^3 = -8$ . Every real number (positive, negative, or zero) has exactly one real cube root.

How do I know if a large number is a perfect cube without full factorisation? Use the unit digit trick first — the cube root’s unit digit is uniquely determined by the number’s unit digit. Then estimate using the cube table (which cube of a two-digit number does this fall between?). This narrows it down to one candidate to verify.

Why does the prime factorisation method work? Because $(a \times b)^3 = a^3 \times b^3$ . If every prime appears three times (i.e., is a perfect cube itself), the whole product is a perfect cube, and we can take the cube root of each prime group independently.

What’s the difference between “multiply to make perfect cube” and “divide to make perfect cube”? Same logic, different direction. In both cases, you want every prime to appear in a multiple of 3. If you can add factors (multiply), you complete incomplete triplets. If you can only remove factors (divide), you eliminate incomplete triplets entirely.

Is the cube root always an integer? No — only when the original number is a perfect cube. $\sqrt[3]{10}$ is an irrational number (approximately 2.154). At Class 8 level, CBSE only tests perfect cubes.

How is this chapter used later in maths? Cube roots appear directly in Class 9 (irrational numbers, number line), Class 10 (real numbers), and are foundational for Class 11 and 12 polynomial functions. Getting comfortable with perfect cube recognition now pays off significantly.

What Are Cubes and Cube Roots?

Key Terms and Definitions

First 10 Perfect Cubes — Memorise These

Core Concepts

Properties of Cubes

The Sum of Consecutive Odd Numbers Pattern

Finding Cube Roots by Prime Factorisation

Solved Examples

Example 1 — Easy (CBSE Level): Is 216 a perfect cube?

Example 2 — Easy (CBSE Level): Find the smallest number by which 392 must be multiplied to make it a perfect cube.

Example 3 — Medium (CBSE Level): Find the smallest number by which 2916 must be divided to make it a perfect cube.

Example 4 — Medium: Find the cube root of −13824-13824−13824.

Example 5 — Hard: Find the cube root of −5122197\frac{-512}{2197}2197−512​.

Exam-Specific Tips

Common Mistakes to Avoid

Practice Questions

Frequently Asked Questions

Practice Questions

Example 4 — Medium: Find the cube root of $-13824$ .

Example 5 — Hard: Find the cube root of $\frac{-512}{2197}$ .