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6.2.2 - Smallest multiple that is a perfect cube

Interactive Audio Lesson

Listen to a student-teacher conversation explaining the topic in a relatable way.

Understanding Perfect Cubes

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Teacher
Teacher

Today, we are diving into what makes a perfect cube. Can anyone tell me an example of a perfect cube?

Student 1
Student 1

Is 27 a perfect cube?

Teacher
Teacher

Great! Yes, 27 is 3 cubed (3 × 3 × 3). Can someone explain why 9 is not a perfect cube?

Student 2
Student 2

Because there is no whole number that can be multiplied by itself three times to get 9.

Teacher
Teacher

Exactly! Remember, a perfect cube results from multiplying a number by itself three times. Let's jot down the cubes of numbers from 1 to 10 on the board.

Student 3
Student 3

Is 2 cubed equal to 8?

Teacher
Teacher

Correct! So, what do we think about the number 32? Is it a perfect cube?

Student 4
Student 4

No! Because its cube root is not a whole number, right?

Teacher
Teacher

Exactly. Now, let’s summarize: Perfect cubes must be expressible as n³ where n is a whole number.

Finding the Smallest Multiple to form a Perfect Cube

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Teacher
Teacher

Now, let’s explore how to find the smallest multiple needed to convert a number into a perfect cube. For instance, we start with 392. What is its prime factorization?

Student 1
Student 1

392 breaks down to 2 × 2 × 2 × 7 × 7.

Teacher
Teacher

Exactly! We see that we have three 2s, but only two 7s. What do we need to make it a perfect cube?

Student 2
Student 2

We need one more 7!

Teacher
Teacher

Yes! So, if we multiply 392 by 7, what do we get?

Student 3
Student 3

2744, which is a perfect cube since it equals 14 cubed.

Teacher
Teacher

Correct! It is essential to identify missing primes and balance their powers to form a perfect cube.

Dividing to Achieve a Perfect Cube

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Teacher
Teacher

Let’s now look at a number we might need to divide to form a perfect cube. For example, 53240. Who can tell me its prime factors?

Student 4
Student 4

I believe it breaks down to 2 × 2 × 2 × 11 × 11 × 11 × 5.

Teacher
Teacher

Good job! What can we conclude from here?

Student 1
Student 1

It seems we have three 11s, but only one 5.

Teacher
Teacher

Very observant! To make 53240 a perfect cube, what would be our action?

Student 2
Student 2

We could divide by 5 to remove the unfit factor!

Teacher
Teacher

Exactly! And doing this gives us a perfect cube of 10648. Let’s wrap up our conversation by summarizing what we learned today regarding converting numbers to perfect cubes.

Introduction & Overview

Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.

Quick Overview

This section explores the concept of perfect cubes and how to determine the smallest multiple of a number that can become a perfect cube.

Standard

In this section, students learn how to analyze the prime factorization of numbers to identify whether they are perfect cubes. The discussion includes examples and problem-solving strategies to find the smallest natural number necessary to convert a number into a perfect cube.

Detailed

Smallest Multiple that is a Perfect Cube

This section covers how to determine the smallest multiple of a number that is a perfect cube, through practical examples and explanations. A perfect cube is defined as a number that can be expressed as the cube of a whole number (e.g., 1, 8, 27, etc.). In the context of the examples, students learn to factor numbers into their prime factors, check their powers, and determine what is needed to achieve the cubic form.

The volumes of cuboids made with plasticine spark a discussion on how many such cuboids would be required to form a perfect cube. Several examples illustrate how to identify if a number is a perfect cube, and if not, how to find the smallest natural number that can be multiplied to yield a perfect cube. Key examples include the factorizations of numbers like 392, 53240, and 1188 to illustrate how to derive the necessary multipliers or divisors to convert them into perfect cubes. This forms a basis for understanding cubes, cube roots, and their significance in mathematics.

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Audio Book

Dive deep into the subject with an immersive audiobook experience.

Volume of a Cuboid

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Raj made a cuboid of plasticine. Length, breadth and height of the cuboid are 15 cm, 30 cm, 15 cm respectively. Anu asks how many such cuboids will she need to make a perfect cube? Can you tell?

Detailed Explanation

In this chunk, we discuss the dimensions of a cuboid – specifically, its length, breadth, and height. A cuboid can be visualized as a box with three dimensions. The volume of a cuboid is calculated by multiplying these three dimensions. For example, the volume is derived as Volume = Length × Breadth × Height, which in this case equals 15 cm * 30 cm * 15 cm.

Examples & Analogies

Consider packing boxes for a shipping company. Each box has a specific size, like Raj's cuboid. Just as the shipping company needs to understand the volume of each box, Raj needs to know how many cuboids will fit into a perfect cube for his project.

Prime Factorization of Volume

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Raj said, Volume of cuboid is 15 × 30 × 15 = 3 × 5 × 2 × 3 × 5 × 3 × 5 = 2 × 3 × 3 × 3 × 5 × 5 × 5.

Detailed Explanation

Here, Raj calculates the volume of his cuboid, which involves breaking down the dimensions into their prime factors. This mathematical technique is valuable because it allows us to express the number in terms of its prime factors, making it easier to determine if the volume is a perfect cube.

Examples & Analogies

Imagine you are organizing items into smaller boxes. Each box's contents can be referred to by factoring them down to their simplest elements. This helps you see how many boxes fit into a larger storage unit, similar to how Raj is trying to see how his cuboids fit into a perfect cube.

Determining the Missing Factors

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Since there is only one 2 in the prime factorisation. So we need 2 × 2, i.e., 4 to make it a perfect cube. Therefore, we need 4 such cuboids to make a cube.

Detailed Explanation

To transform the volume of the cuboid into a perfect cube, we require every prime factor to appear in multiples of three. Since the factor '2' appears only once, we will need to add two more factors of '2' to form a full triplet. Therefore, we conclude that four cuboids will provide the necessary factors to create a perfect cube.

Examples & Analogies

Think of a recipe that calls for groups of three ingredients. If you only have one ingredient, you can't complete the recipe until you gather more. Similarly, Raj needs to gather enough cuboids to complete his perfect cube 'recipe'.

Checking if 392 is a Perfect Cube

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Example 2: Is 392 a perfect cube? If not, find the smallest natural number by which 392 must be multiplied so that the product is a perfect cube. Solution: 392 = 2 × 2 × 2 × 7 × 7.

Detailed Explanation

In this chunk, we analyze whether the number 392 is a perfect cube. By factorizing it, we find that it can be represented by prime factors. To determine if a number is a perfect cube, we look for each prime factor to occur in groups of three.

Examples & Analogies

Imagine a sports team where each player has a role – if there aren't enough players in each role, the team can't function. Similarly, for 392 to be a perfect cube, it needs the right number of each prime factor (players) to be organized in the correct group (three at a time).

Finding the Necessary Multiplier

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To make its a cube, we need one more 7. In that case 392 × 7 = 2 × 2 × 2 × 7 × 7 × 7 = 2744 which is a perfect cube. Hence the smallest natural number by which 392 should be multiplied to make a perfect cube is 7.

Detailed Explanation

After determining that 392 is not a perfect cube, we calculate that multiplying it by 7 introduces another factor of 7, thereby allowing every prime number to occur in groups of three. The result, 2744, is a perfect cube. Therefore, 7 is identified as the smallest natural number needed to convert 392 into a perfect cube.

Examples & Analogies

Think of adding more chairs to a dining table setting. If you need all sets of three for a proper arrangement, then adding one more chair (or factor) completes your setup. Here, multiplying by 7 completes the necessary setup for a perfect cube.

Definitions & Key Concepts

Learn essential terms and foundational ideas that form the basis of the topic.

Key Concepts

  • Perfect Cubes: Must be expressible as n³, where n is a whole number.

  • Prime Factorization: Breakdown of a number into prime factors is essential for identifying perfect cubes.

  • Smallest Multiple: To convert a number into a perfect cube, add the required factors based on prime factorization.

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

  • Example of 392: Factorizes to 2 × 2 × 2 × 7 × 7; needs one more 7 to form 2744.

  • Example of 53240: Factorizes to 2 × 2 × 2 × 11 × 11 × 11 × 5; divide by 5 to achieve 10648.

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🎵 Rhymes Time

  • To cube a number, multiply three times, one, eight, and twenty-seven—a rhythm in rhymes.

📖 Fascinating Stories

  • Once a builder wanted to create a perfect cube from his small bricks but realized he needed the same quantities, so he counted his odd bricks and found the missing pieces to fit perfectly!

🧠 Other Memory Gems

  • CUBED: Count, Understand, Balance, Evaluate, Determine—a reminder to check all factor counts.

🎯 Super Acronyms

CUBE means

  • C: - Cube
  • U: - Understand factors
  • B: - Balance counts
  • E: - Ensure its perfect!

Flash Cards

Review key concepts with flashcards.

Glossary of Terms

Review the Definitions for terms.

  • Term: Perfect Cube

    Definition:

    A number that can be expressed as the cube of an integer, such as 1, 8, 27, etc.

  • Term: Prime Factorization

    Definition:

    The process of breaking down a number into its basic building blocks, or prime numbers.

  • Term: Volume

    Definition:

    The amount of space occupied by a three-dimensional object, often calculated by multiplying length, breadth, and height.