Mastering Cube and Cube Roots: NCERT Ex 7.2 Class 8 Maths

Welcome, Class 8 Maths explorers! You've already mastered squares and square roots. Now, get ready to dive into the equally fascinating world of 'Cubes and Cube Roots'. This chapter, specifically focusing on Exercise 7.2 from your NCERT textbook, is all about understanding what a cube number is and how to find its cube root. Imagine building a perfect cube shape using small, identical blocks – the total number of blocks would be a cube number! Finding its cube root is like figuring out the length of one side of that perfect cube. This concept is vital, not just for exams, but also for understanding volumes in geometry. By the end of this page, you'll be an expert at finding cube roots using the powerful prime factorization method, preparing you to confidently solve every problem in 'Cube and Cube Roots Ex 7 2' and beyond.

What are Cube Numbers and Cube Roots?

A cube number, also known as a perfect cube, is a number you get by multiplying a whole number by itself three times. For example, if you take the number 2 and multiply it by itself three times (2 × 2 × 2), you get 8. So, 8 is a perfect cube. Similarly, 3 × 3 × 3 = 27, which means 27 is also a perfect cube. Think of it like a 3D block: if a cube has sides of length 2 units, its volume is 2³ = 8 cubic units.

The cube root of a number is the value that, when multiplied by itself three times, gives you the original number. It's the opposite operation of cubing a number. The special symbol for cube root is \(\sqrt[3]{}\). So, \(\sqrt[3]{8} = 2\) because 2 x 2 x 2 = 8. And \(\sqrt[3]{27} = 3\) because 3 x 3 x 3 = 27. Understanding cube numbers and their roots is fundamental for solving problems in 'Cube and Cube Roots Ex 7 2'. While small cube roots might be easy to remember, for larger numbers, we need a systematic approach, which we'll explore next.

Finding Cube Roots Using Prime Factorization (Step-by-Step)

  1. Step 1: Prime Factorize the Number — Start by finding the prime factors of the given number. Divide the number by the smallest prime number (2, 3, 5, 7, etc.) that divides it evenly, and continue this process until you are left with only prime numbers as factors.
  2. Step 2: Group Prime Factors in Triplets — Once you have all the prime factors, group them into sets of three identical factors. Each group of three should consist of the same prime number. For example, if you have 2 × 2 × 2 × 3 × 3 × 3, you would make one group of (2 × 2 × 2) and another group of (3 × 3 × 3).
  3. Step 3: Take One Factor from Each Triplet — From each group of three identical prime factors, take out just one of those prime factors. So, from (2 × 2 × 2), you take out a single '2'. From (3 × 3 × 3), you take out a single '3'.
  4. Step 4: Multiply the Chosen Factors — Multiply all the single prime factors you took out from each triplet. The product will be the cube root of the original number. If any prime factor is left over and cannot form a triplet, then the original number is not a perfect cube.

Worked Examples: Finding Cube Roots

  • Example 1: Find the cube root of 512. Step 1: Prime Factorize 512 512 ÷ 2 = 256 256 ÷ 2 = 128 128 ÷ 2 = 64 64 ÷ 2 = 32 32 ÷ 2 = 16 16 ÷ 2 = 8 8 ÷ 2 = 4 4 ÷ 2 = 2 2 ÷ 2 = 1 So, the prime factorization of 512 is 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2. Step 2: Group Prime Factors in Triplets We have nine '2's. We can form three triplets of 2s: (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2) Step 3: Take One Factor from Each Triplet From each triplet, we take one '2'. So, we get 2, 2, and 2. Step 4: Multiply the Chosen Factors 2 × 2 × 2 = 8 Therefore, the cube root of 512 is 8. (\(\sqrt[3]{512} = 8\))
  • Example 2: Find the cube root of 13824. Step 1: Prime Factorize 13824 13824 ÷ 2 = 6912 6912 ÷ 2 = 3456 3456 ÷ 2 = 1728 1728 ÷ 2 = 864 864 ÷ 2 = 432 432 ÷ 2 = 216 216 ÷ 2 = 108 108 ÷ 2 = 54 54 ÷ 2 = 27 27 ÷ 3 = 9 9 ÷ 3 = 3 3 ÷ 3 = 1 So, 13824 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3. Step 2: Group Prime Factors in Triplets (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2) × (3 × 3 × 3) Step 3: Take One Factor from Each Triplet From the 2s, we get 2, 2, 2. From the 3s, we get 3. Step 4: Multiply the Chosen Factors 2 × 2 × 2 × 3 = 8 × 3 = 24 Therefore, the cube root of 13824 is 24. (\(\sqrt[3]{13824} = 24\))

Exam Tip: Avoiding Common Mistakes in Cube Roots

When finding cube roots using the prime factorization method, students often make a few common errors. Firstly, make sure you perform the prime factorization correctly and completely; missing a prime factor or incorrectly dividing can lead to wrong answers. Secondly, and very importantly, remember to group the factors in triplets (sets of three identical factors). If you group them in pairs (like for square roots), your answer will be incorrect. Thirdly, always double-check your multiplication in the final step. Lastly, for problems asking if a number is a perfect cube, if any prime factor is left over and cannot form a complete triplet, then the number is not a perfect cube.

Practice Questions with Solutions

  • Q: Find the cube root of 729 by prime factorization method. A: Step 1: Prime factorize 729. 729 ÷ 3 = 243 243 ÷ 3 = 81 81 ÷ 3 = 27 27 ÷ 3 = 9 9 ÷ 3 = 3 3 ÷ 3 = 1 So, 729 = 3 × 3 × 3 × 3 × 3 × 3. Step 2: Group prime factors in triplets. (3 × 3 × 3) × (3 × 3 × 3) Step 3: Take one factor from each triplet. We get 3 and 3. Step 4: Multiply the chosen factors. 3 × 3 = 9 Final answer: The cube root of 729 is 9.
  • Q: Is 64000 a perfect cube? If yes, find its cube root. A: Step 1: Prime factorize 64000. 64000 = 64 × 1000 64 = 2 × 2 × 2 × 2 × 2 × 2 (2⁶) 1000 = 10 × 10 × 10 = (2 × 5) × (2 × 5) × (2 × 5) = 2 × 2 × 2 × 5 × 5 × 5 (2³ × 5³) So, 64000 = (2⁶) × (2³ × 5³) = 2⁹ × 5³ Step 2: Group prime factors in triplets. (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2) × (5 × 5 × 5) All factors form complete triplets. Step 3: Take one factor from each triplet. We get 2, 2, 2, and 5. Step 4: Multiply the chosen factors. 2 × 2 × 2 × 5 = 8 × 5 = 40 Final answer: Yes, 64000 is a perfect cube, and its cube root is 40.
  • Q: Find the smallest number by which 256 must be multiplied to obtain a perfect cube. A: Step 1: Prime factorize 256. 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 (2⁸) Step 2: Group prime factors in triplets. (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2) We have two triplets of 2s, but the last group (2 × 2) only has two '2's. To make it a triplet, it needs one more '2'. Step 3: Identify the missing factor. One '2' is missing to complete the last triplet. Step 4: Determine the smallest number to multiply. The smallest number is 2. Final answer: 256 must be multiplied by 2 to obtain a perfect cube (256 × 2 = 512, and \(\sqrt[3]{512} = 8\)).
  • Q: Find the smallest number by which 81 must be divided to obtain a perfect cube. A: Step 1: Prime factorize 81. 81 = 3 × 3 × 3 × 3 (3⁴) Step 2: Group prime factors in triplets. (3 × 3 × 3) × (3) We have one triplet of 3s, and one '3' is left over. Step 3: Identify the excess factor. The excess factor is 3. Step 4: Determine the smallest number to divide. To make 81 a perfect cube, we must divide by the excess factor, which is 3. Final answer: 81 must be divided by 3 to obtain a perfect cube (81 ÷ 3 = 27, and \(\sqrt[3]{27} = 3\)).

Frequently Asked Questions

What is a perfect cube?

A perfect cube is a number that can be obtained by multiplying an integer by itself three times. For example, 64 is a perfect cube because 4 × 4 × 4 = 64. These numbers are also called cube numbers.

How do I find the cube root of a large number?

The most effective method for finding the cube root of a large number is prime factorization. You break the number down into its prime factors, group identical factors into triplets, and then multiply one factor from each triplet to get the cube root.

What is the difference between a square root and a cube root?

A square root (√) finds a number that, when multiplied by itself *twice*, gives the original number (e.g., √25 = 5). A cube root (√³) finds a number that, when multiplied by itself *three* times, gives the original number (e.g., \(\sqrt[3]{125} = 5\)). The small '3' in the cube root symbol is key.

Can a negative number have a cube root?

Yes, unlike square roots, a negative number can have a real cube root. For instance, \(\sqrt[3]{-8} = -2\), because (-2) × (-2) × (-2) = -8. The sign of the cube root will be the same as the sign of the original number.