Welcome to our complete guide for your Class 8 Mathematics textbook! In this post, we provide clear, step-by-step solutions for the "Figure It Out" exercise from Chapter 1, focusing on Perfect Squares and Perfect Cubes. Understanding concepts like prime factorization for perfect cubes and comparing the growth rates of squares and cubes is crucial for mastering this chapter.
We will break down each problem from the exercise on Page 16, ensuring our explanations are easy to follow and help you excel in your studies.
Q1: Finding the Cube Root of Large Numbers (27000 & 10648)
Finding the cube root of a number n, denoted as ∛n, means finding a number that, when multiplied by itself three times, equals n.
Calculation for 27000:
We can break down 27000 into factors we recognize.
Since we know 3³ = 27 and 10³ = 1000, we can write:
27000 = 27 × 1000 = 3³ × 10³
Using the exponent rule (a × b)ᵐ = aᵐ × bᵐ, we get:
(3 × 10)³ = 30³
Therefore, the cube root of 27000 is 30.
Answer: ∛27000 = 30
Calculation for 10648:
For a number like 10648, prime factorization is the most reliable method. Let's find the prime factors:
10648 = 2 × 5324
= 2 × 2 × 2662
= 2 × 2 × 2 × 1331
Now, we need to factor 1331. We can test small prime numbers. We find it is divisible by 11:
1331 = 11 × 121
= 11 × 11 × 11
So, the complete prime factorization of 10648 is:
10648 = 2 × 2 × 2 × 11 × 11 × 11 = 2³ × 11³
Combining these terms, we have:
(2 × 11)³ = 22³
Therefore, the cube root of 10648 is 22.
Answer: ∛10648 = 22
Q2: How to Make 1323 a Perfect Cube
To make a number a perfect cube, we first find its prime factorization. Then, we ensure every prime factor appears in a group of three (a triplet).
Step 1: Find the Prime Factorization of 1323
Let's break down 1323 into its prime factors:
1323 = 3 × 441
We know that 441 = 21², and 21 = 3 × 7. So:
1323 = 3 × (21 × 21) = 3 × (3 × 7) × (3 × 7)
Grouping the factors together, we get:
1323 = (3 × 3 × 3) × (7 × 7) = 3³ × 7²
Step 2: Identify the Missing Factor
Looking at the factorization 3³ × 7², we see that the factor 3 forms a complete triplet (3³). However, the factor 7 only appears twice (7²). To make it a perfect cube, we need one more 7 to create a triplet (7³).
Answer: You must multiply 1323 by 7 to make it a perfect cube.
The resulting perfect cube would be 1323 × 7 = 9261, and its cube root is ∛9261 = 21.
Q3: True or False? Testing Properties of Cubes
This question tests your understanding of the properties of perfect cubes, such as their parity (even/odd) and digit patterns.
| Statement | True/False | Reasoning |
|---|---|---|
| (i) The cube of any odd number is even. | False | The cube of an odd number is always odd. For example, 3³ = 27 and 5³ = 125, both of which are odd. |
| (ii) There is no perfect cube that ends with 8. | False | Perfect cubes can end in 8. For example, 2³ = 8. Any number ending in 2 will have a cube that ends in 8 (e.g., 12³ = 1728). |
| (iii) The cube of a 2-digit number may be a 3-digit number. | False | The smallest 2-digit number is 10. Its cube is 10³ = 1000, which is a 4-digit number. All other 2-digit cubes will be larger. |
| (iv) The cube of a 2-digit number may have seven or more digits. | False | The largest 2-digit number is 99. Its cube is 99³ = 970,299, which is a 6-digit number. No 2-digit cube can reach 7 digits. |
| (v) Cube numbers have an odd number of factors. | False | This property is true for perfect squares, but not for perfect cubes. For example, the factors of 8 (2³) are 1, 2, 4, and 8—a total of four factors, which is an even number. |
Q4: Guessing Cube Roots Without Factorization
You can often find the cube root of a perfect cube by recognizing patterns in its digits.
Method:
- Check the Units Digit: The last digit of a cube's root is determined by the last digit of the cube itself.
- Ends in 1 → Root ends in 1
- Ends in 8 → Root ends in 2
- Ends in 7 → Root ends in 3
- Ends in 4 → Root ends in 4
- Ends in 5 → Root ends in 5
- Ends in 6 → Root ends in 6
- Ends in 3 → Root ends in 7
- Ends in 2 → Root ends in 8
- Ends in 9 → Root ends in 9
- Ends in 0 → Root ends in 0
- Estimate the Range: Find two consecutive perfect cubes (like 10³ and 20³) that your number lies between. This will tell you the first digit of the root.
Guessing the Roots:
• 1331: It ends in 1, so the root must end in 1. It is between 10³ = 1000 and 20³ = 8000.
Guess: 11. (Check: 11³ = 1331. Correct!)
• 4913: It ends in 3, so the root must end in 7. It is between 10³ = 1000 and 20³ = 8000.
Guess: 17. (Check: 17³ = 4913. Correct!)
• 12167: It ends in 7, so the root must end in 3. It is between 20³ = 8000 and 30³ = 27000.
Guess: 23. (Check: 23³ = 12167. Correct!)
• 32768: It ends in 8, so the root must end in 2. It is between 30³ = 27000 and 40³ = 64000.
Guess: 32. (Check: 32³ = 32768. Correct!)
Q5: Comparing the Difference of Squares vs. Cubes
This problem asks which of the four given expressions has the largest value. We can use algebraic identities for the difference of squares and cubes to solve this efficiently.
- Difference of Squares: a² − b² = (a − b)(a + b)
- Difference of Cubes: a³ − b³ = (a − b)(a² + ab + b²)
In each option, the two numbers are consecutive, so a − b = 1. This means we only need to compare the second part of each formula.
| Option | Expression | Simplified Comparison | Value |
|---|---|---|---|
| (i) | 67³ − 66³ | 67² + (67 × 66) + 66² | 13,267 |
| (ii) | 43³ − 42³ | 43² + (43 × 42) + 42² | 5,419 |
| (iii) | 67² − 66² | 67 + 66 | 133 |
| (iv) | 43² − 42² | 43 + 42 | 85 |
Conclusion:
As you can see from the values, the difference between consecutive cubes grows much faster than the difference between consecutive squares. The largest value comes from the expression with the highest power (cubes) and the largest base numbers (67 and 66).
Answer: The greatest value is (i) 67³ − 66³.
Continue Your Grade 8 Maths Journey!
Congratulations on working through these challenging problems! These exercises build a strong foundation for understanding exponents and roots. If you are ready to move on, check out the next section of your curriculum.
➡️ Click here for Chapter 1 Solutions Page 10
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