NCERT Class 8 Maths: Cube and Cube Roots

Welcome, young mathematicians! In geometry, you have learned that a cube is a 3-dimensional solid shape with all sides equal. In arithmetic, the concept of a 'cube' of a number is directly linked to this shape. When you multiply a number by itself three times, you get its cube. For instance, the cube of 2 is 2 times 2 times 2, which equals 8. Conversely, finding the 'cube root' is the reverse operation—discovering which number multiplied by itself thrice yields the given value. Mastering this chapter is crucial as it lays the foundation for higher-grade algebra, solid geometry, and real-world mensuration problems. In this guide, your YoLearn AI Tutor will walk you through finding cubes, identifying perfect cubes, and finding cube roots using the prime factorization method with step-by-step clarity.

What are Cubes and Perfect Cubes?

In CBSE Class 8, a perfect cube (or a cube number) is defined as a natural number that is the product of three identical natural numbers. Mathematically, if $y = x \times x \times x = x^3$, then $y$ is the cube of $x$. For example, $1^3 = 1$, $2^3 = 8$, $3^3 = 27$, and $4^3 = 64$. Here, 1, 8, 27, and 64 are perfect cubes. If we represent this visually, a perfect cube number of identical small unit blocks can always be arranged to form one large, solid cube. Note that most numbers are not perfect cubes; for instance, 9 is a perfect square ($3 \times 3$) but not a perfect cube because no natural number multiplied by itself three times equals 9.

How to Find Cube Roots Using Prime Factorization

  1. Step 1: Resolve the number into prime factors — Perform prime division on the given number until you reach 1, writing down all the prime factors.
  2. Step 2: Group the factors into triplets — Group the identical prime factors into sets of three (triplets). For a perfect cube, every prime factor must appear in complete triplets with none left over.
  3. Step 3: Select one factor from each triplet — From each group of three identical factors, choose one representative prime factor.
  4. Step 4: Multiply the chosen factors — Multiply these selected factors together. The resulting product is the cube root of the given number.

Key Properties of Cube Numbers

  • Cubes of even numbers are always even. For example, $2^3 = 8$ and $6^3 = 216$.
  • Cubes of odd numbers are always odd. For example, $3^3 = 27$ and $5^3 = 125$.
  • The cube of a negative number is always negative. For example, $(-3)^3 = -3 \times -3 \times -3 = -27$.
  • If a number ends with a digit 'a', its cube ends with a specific unit digit. Numbers ending in 1, 4, 5, 6, 9, and 0 have cubes ending in the same digit. Cubes of numbers ending in 2 end in 8, and vice versa. Cubes of numbers ending in 3 end in 7, and vice versa.

Avoid This Common Board Exam Mistake

A very common error Class 8 students make is confusing Square Roots with Cube Roots. When writing the symbol for a cube root, always include the small index '3' inside the radical hook, like this: $\sqrt[3]{x}$. Writing just $\sqrt{x}$ represents the square root. Also, remember that during prime factorization, square roots require making pairs of two, while cube roots require making triplets of three identical factors. Double-check your groupings before writing your final answer!

Practice Questions with Solutions

  • Q: Find the cube root of 216 using prime factorization. A: Step 1: Find the prime factors of 216. $216 = 2 \times 2 \times 2 \times 3 \times 3 \times 3$ Step 2: Group the prime factors into triplets. $216 = (2 \times 2 \times 2) \times (3 \times 3 \times 3)$ Step 3: Take one factor from each triplet. $\sqrt[3]{216} = 2 \times 3$ Step 4: Multiply the chosen factors. $2 \times 3 = 6$ Final answer: The cube root of 216 is 6.
  • Q: Is 392 a perfect cube? If not, find the smallest natural number by which it must be multiplied so that the product is a perfect cube. A: Step 1: Find the prime factors of 392. $392 = 2 \times 196 = 2 \times 2 \times 98 = 2 \times 2 \times 2 \times 7 \times 7$ Step 2: Group the prime factors into triplets. $392 = (2 \times 2 \times 2) \times 7 \times 7$ Step 3: Identify missing factors. The prime factor 7 does not form a triplet; it appears only twice. Step 4: Determine the multiplier. To complete the triplet of 7s, we need to multiply by one more 7. $392 \times 7 = (2 \times 2 \times 2) \times (7 \times 7 \times 7) = 2744$ (which is $14^3$) Final answer: 392 is not a perfect cube. The smallest natural number by which 392 must be multiplied is 7.
  • Q: Find the smallest number by which 81 must be divided so that the quotient is a perfect cube. A: Step 1: Prime factorize 81. $81 = 3 \times 3 \times 3 \times 3$ Step 2: Group the prime factors into triplets. $81 = (3 \times 3 \times 3) \times 3$ Step 3: Identify the extra factors. The triplet $(3 \times 3 \times 3)$ is complete, but there is one extra '3' left over. Step 4: Divide the number by this extra factor to make it a perfect cube. $81 \div 3 = 27$ (which is $3^3$) Final answer: The smallest number by which 81 must be divided is 3.
  • Q: Find the cube root of 512 using the prime factorization method. A: Step 1: Prime factorize 512. $512 = 2 \times 256 = 2 \times 2 \times 128 = 2 \times 2 \times 2 \times 64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$ Step 2: Group the prime factors into triplets. $512 = (2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (2 \times 2 \times 2)$ Step 3: Take one factor from each triplet and multiply. $\sqrt[3]{512} = 2 \times 2 \times 2 = 8$ Final answer: The cube root of 512 is 8.

Frequently Asked Questions

What is a perfect cube?

A perfect cube is an integer that can be written as the product of three equal integers. For example, 27 is a perfect cube because it equals $3 \times 3 \times 3$.

Can a perfect cube end with two zeroes?

No, a perfect cube can never end with exactly two zeroes. Because a cube is multiplied three times, trailing zeroes will always appear in multiples of three (e.g., 1000 has three zeroes, 1,000,000 has six zeroes).

How do you find the cube root of a negative integer?

The cube root of a negative integer is always negative. To find it, calculate the cube root of its positive absolute value first, and then place a negative sign in front of the result.