Cubes And Cube Roots - CBSE Class 8 Maths

Welcome, Class 8 students! In this exciting chapter, "Cubes And Cube Roots", we're going to dive into a fascinating area of numbers. You might already know about squares and square roots, where a number is multiplied by itself twice. Now, imagine multiplying a number by itself three times! That's what a cube number is, and finding its cube root is like reversing that process. Understanding cubes and cube roots isn't just for textbooks; it helps us calculate the volume of cubic objects and is a fundamental skill in higher mathematics. By the end of this page, you'll be able to identify perfect cube numbers, find the cube root of numbers using the prime factorization method, and solve a variety of problems confidently. Let's unlock the power of these special numbers together!

What Are Cube Numbers?

A cube number, also known as a perfect cube, is a number obtained by multiplying an integer by itself three times. For example, if you have a number 'x', its cube is x × x × x, which is written as x³. Think about a physical cube – it has equal length, width, and height. If each side of a cube is 'x' units long, its volume is x³ cubic units. This is why these numbers are called "cubes"!

Let's look at some examples of cube numbers:

  • 1³ = 1 × 1 × 1 = 1
  • 2³ = 2 × 2 × 2 = 8
  • 3³ = 3 × 3 × 3 = 27
  • 4³ = 4 × 4 × 4 = 64
  • 5³ = 5 × 5 × 5 = 125

Notice how quickly cube numbers grow! Understanding these basic cubes will be very helpful as we move to finding cube roots. Also, observe the last digit of these cube numbers: 1 ends in 1, 8 ends in 8, 27 ends in 7, 64 ends in 4, 125 ends in 5. This pattern of ending digits is a useful property we'll explore further.

Finding Cube Roots by Prime Factorization

  1. Step 1: Perform Prime Factorization — Break down the given number into its prime factors. This means expressing the number as a product of prime numbers only. Remember, a prime number is a number greater than 1 that has only two factors: 1 and itself (e.g., 2, 3, 5, 7, 11, etc.).
  2. Step 2: Group Factors in Triplets — Once you have all the prime factors, group identical factors in sets of three (triplets). For a number to be a perfect cube, all its prime factors must be able to form complete triplets. If even one prime factor is left ungrouped or doesn't form a complete triplet, the number is not a perfect cube.
  3. Step 3: Take One Factor from Each Triplet — From each group of three identical prime factors, take out just one factor. For example, if you have (2 × 2 × 2), you take out one '2'.
  4. Step 4: Multiply the Chosen Factors — Multiply all the factors you took out from each triplet. The product will be the cube root of the original number. This is the number that, when multiplied by itself three times, gives you the original number.

Worked Example: Finding the Cube Root of 1728

  • Let's find the cube root of 1728 using the prime factorization method.

    Step 1: Perform Prime Factorization of 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, 1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3

    Step 2: Group Factors in Triplets
    We group the identical prime factors into sets of three:
    1728 = (2 × 2 × 2) × (2 × 2 × 2) × (3 × 3 × 3)

    Step 3: Take One Factor from Each Triplet
    From the first triplet of 2s, we take one 2.
    From the second triplet of 2s, we take one 2.
    From the triplet of 3s, we take one 3.

    Step 4: Multiply the Chosen Factors
    Multiply these chosen factors: 2 × 2 × 3 = 12.

    Final Answer: The cube root of 1728 is 12. (We can check: 12 × 12 × 12 = 144 × 12 = 1728)

Exam Tip: Quick Checks for Perfect Cubes

When you're trying to determine if a number is a perfect cube or calculating its cube root, keep these points in mind:

  1. Last Digit Pattern: The last digit of a perfect cube can tell you the last digit of its cube root. For example:
  • Numbers ending in 0, 1, 4, 5, 6, 9 will have cube roots ending in 0, 1, 4, 5, 6, 9 respectively.
  • Numbers ending in 2 will have cube roots ending in 8 (and vice-versa).
  • Numbers ending in 3 will have cube roots ending in 7 (and vice-versa).

This pattern is unique to cubes and can help you verify your answers or even estimate cube roots for larger numbers quickly.

  1. Prime Factorization is Key: Always ensure that all prime factors appear in groups of three. If a factor appears once, twice, four times, five times, etc., the number is not a perfect cube. For instance, if you get 2 x 2 x 2 x 3 x 3, the number is not a perfect cube because the '3's do not form a complete triplet. To make it a perfect cube, you would need to multiply it by another '3'.

Practice Questions with Solutions

  • Q: Find the cube root of 512.
    A: Step 1: Prime factorization of 512:
    512 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
    (There are nine '2's)
    Step 2: Group factors in triplets:
    512 = (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2)
    Step 3: Take one factor from each triplet:
    2 × 2 × 2
    Step 4: Multiply the chosen factors:
    2 × 2 × 2 = 8
    Final answer: The cube root of 512 is 8.

  • Q: Is 343 a perfect cube? If yes, find its cube root.
    A: Step 1: Prime factorization of 343:
    343 ÷ 7 = 49
    49 ÷ 7 = 7
    7 ÷ 7 = 1
    So, 343 = 7 × 7 × 7
    Step 2: Group factors in triplets:
    343 = (7 × 7 × 7)
    Step 3: All prime factors form complete triplets. Yes, 343 is a perfect cube.
    Step 4: Take one factor from the triplet: 7
    Final answer: Yes, 343 is a perfect cube. Its cube root is 7.

  • Q: What is the smallest number by which 256 must be multiplied to get a perfect cube?
    A: Step 1: Prime factorization of 256:
    256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
    (There are eight '2's)
    Step 2: Group factors in triplets:
    256 = (2 × 2 × 2) × (2 × 2 × 2) × 2 × 2
    We have two complete triplets of '2's, and two '2's remaining.
    Step 3: To make it a perfect cube, the remaining factors (2 × 2) need one more '2' to form a triplet.
    Step 4: The smallest number to multiply is 2.
    Final answer: 256 must be multiplied by 2 to get a perfect cube (which would be 512).

  • Q: Find the cube root of 1331.
    A: Step 1: Prime factorization of 1331:
    1331 ÷ 11 = 121
    121 ÷ 11 = 11
    11 ÷ 11 = 1
    So, 1331 = 11 × 11 × 11
    Step 2: Group factors in triplets:
    1331 = (11 × 11 × 11)
    Step 3: Take one factor from the triplet: 11
    Final answer: The cube root of 1331 is 11.

Frequently Asked Questions

What is the main difference between a square number and a cube number?

A square number results from multiplying an integer by itself twice (e.g., 3 × 3 = 9). A cube number results from multiplying an integer by itself three times (e.g., 3 × 3 × 3 = 27). Visually, squares relate to the area of a square, while cubes relate to the volume of a cube.

How can I quickly tell if a number is NOT a perfect cube?

You can quickly check the last digit. Numbers ending in 2, 3, 7, or 8 can be perfect cubes. If a number ends in any other digit (0, 1, 4, 5, 6, 9) it might be a perfect cube. The most reliable method, however, is prime factorization; if any prime factor doesn't form a triplet, it's not a perfect cube.

What is the cube root symbol?

The symbol for a cube root is similar to the square root symbol, but with a small '3' placed above the check mark, like this: ∛. So, ∛x means the cube root of x. For example, ∛27 = 3.

Can negative numbers have cube roots?

Yes, unlike square roots, negative numbers can have real cube roots. For example, the cube root of -8 is -2, because (-2) × (-2) × (-2) = -8. This is because multiplying a negative number an odd number of times results in a negative product.