NCERT Class 8 Maths Cube and Cube Roots Exercise 7.1 Guide

Understanding Perfect Cubes & The Triplet Rule

A natural number is called a perfect cube (or a cube number) if it is the cube of some natural number. For example, 1, 8, 27, and 64 are perfect cubes because they are $1^3$, $2^3$, $3^3$, and $4^3$ respectively.

To identify whether a larger number is a perfect cube, we use its prime factorization. Unlike square numbers where we group prime factors in pairs of two, for perfect cubes, we group the prime factors into triplets (groups of three identical factors). If all prime factors of a number can be grouped into complete triplets with none left over, the number is a perfect cube. If any factor is left without a triplet, then the number is not a perfect cube. This simple rule is the foundation of solving all questions in Exercise 7.1.

How to Make a Non-Perfect Cube a Perfect Cube

1. Perform Prime Factorization — Divide the given number by its smallest prime factors (2, 3, 5, 7...) repeatedly until you reach 1. Write down the complete prime product.
2. Group Factors into Triplets — Arrange the prime factors in groups of three identical numbers (e.g., 2 × 2 × 2, 3 × 3 × 3).
3. Identify Missing or Extra Factors — Look for groups that have fewer than three factors. If you need to make it a perfect cube by MULTIPLICATION, find what factors are missing to complete the triplet. If you need to make it a perfect cube by DIVISION, identify the extra factors that do not form a complete triplet.
4. Calculate the Smallest Number — Multiply the missing factors (for multiplication problems) or the extra factors (for division problems) to get your final required number.

Common Exam Mistakes to Avoid

• The Pairing Trap: Students often confuse square roots with cube roots. Remember, for square roots, you group factors in pairs of 2. For cube roots and perfect cubes, you must group them in triplets of 3.
• Multiplication vs. Division Confusion: In multiplication questions, you find what is missing to complete a triplet. In division questions, you divide by the entirety of the incomplete group to get rid of it. Do not mix up these two methods!
• Neglecting Prime Factorization Order: Always start factorizing with the smallest prime numbers (like 2, then 3, then 5) to keep your calculations clean and error-free.

Practice Questions with Solutions

• Q: Check whether 216 is a perfect cube. A: Step 1: Perform the prime factorization of 216. 216 = 2 × 108 108 = 2 × 54 54 = 2 × 27 27 = 3 × 9 9 = 3 × 3 3 = 3 × 1 So, 216 = 2 × 2 × 2 × 3 × 3 × 3 Step 2: Group the prime factors into triplets. 216 = (2 × 2 × 2) × (3 × 3 × 3) Step 3: Check if any factor is left over. All factors are grouped into complete triplets. No factor is left ungrouped. Final answer: Yes, 216 is a perfect cube.
• Q: Is 128 a perfect cube? If not, find the smallest number by which it must be divided so that the quotient is a perfect cube. A: Step 1: Find the prime factorization of 128. 128 = 2 × 64 64 = 2 × 32 32 = 2 × 16 16 = 2 × 8 8 = 2 × 4 4 = 2 × 2 2 = 2 × 1 So, 128 = 2 × 2 × 2 × 2 × 2 × 2 × 2 Step 2: Group the prime factors into triplets. 128 = (2 × 2 × 2) × (2 × 2 × 2) × 2 Step 3: Analyze the ungrouped factors. We can see that one '2' is left over and does not form a triplet. Therefore, 128 is not a perfect cube. To make it a perfect cube, we must divide 128 by this extra 2. Final answer: The smallest number by which 128 must be divided is 2.
• Q: Find the smallest number by which 72 must be multiplied to obtain a perfect cube. A: Step 1: Prime factorize 72. 72 = 2 × 36 36 = 2 × 18 18 = 2 × 9 9 = 3 × 3 3 = 3 × 1 So, 72 = 2 × 2 × 2 × 3 × 3 Step 2: Group the factors into triplets. 72 = (2 × 2 × 2) × (3 × 3) Step 3: Identify the missing factors. The prime factor 2 forms a complete triplet, but the prime factor 3 occurs only twice. To form a triplet of 3, we need one more factor of 3 (since 3 × 3 × 3 is a triplet). Final answer: The smallest number by which 72 must be multiplied is 3.
• Q: Parikshit makes a cuboid of plasticine of sides 5 cm, 2 cm, 5 cm. How many such cuboids will he need to form a cube? A: Step 1: Understand the volume of the single cuboid. Volume of cuboid = 5 cm × 2 cm × 5 cm = $5^2 × 2^1$ $cm^3$. Step 2: Determine what factors are needed to make the volume a perfect cube. To make it a perfect cube, the power of each prime factor must be a multiple of 3. Currently, we have two 5s ($5^2$) and one 2 ($2^1$). To complete the triplet of 5, we need one more 5. To complete the triplet of 2, we need two more 2s (i.e., 2 × 2). Step 3: Calculate the total multiplier. Multiplier needed = 5 × 2 × 2 = 20. Final answer: Parikshit will need 20 such cuboids to form a perfect cube.

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