Cube Roots Through Prime Factorization-Class-8-Cubes and Cuberoots-Lesson plan-6.7-6.8for AP/TS/NCERT

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Cube Roots Through Prime Factorization

By: M. RajaRao, MSc, MEd

45-Minute Interactive Mathematics Lesson

Hook Activity
4 mins

I Do
8 mins

We Do
16 mins

You Do
8 mins

Assessment
9 mins

📚 Learning Objectives

Period 6.7:

Students will be able to find the Cubes having their cube roots from 1 to 20

Period 6.8:

Students will be able to find the cube root of large cube numbers through the prime factorization method

🎣 Hook Activity (4 Minutes) - Grab Students' Attention

Let's start with an engaging cube visualization to activate prior knowledge!

🧮 Quick Cube Calculator

Enter a number to see its cube and explore the concept:

Try entering a number from 1 to 20!

🤔 Think About This:

If 2³ = 8, then what is ∛8? How can we find cube roots of larger numbers like 729 or 1728?

👨‍🏫 Explicit Teaching - I Do (8 Minutes)

Prime Factorization Method for Cube Roots

📝 Example 1: Find ∛729

Step 1: Prime Factorization using Short Division Method

3 | 729
3 | 243
3 | 81
3 | 27
3 | 9
3 | 3
  | 1

Step 2: 729 = 3 × 3 × 3 × 3 × 3 × 3 = (3 × 3 × 3) × (3 × 3 × 3) = 3³ × 3³

Step 3: Grouping the factors into triplets, we get (3 × 3)³ = 9³

Step 4: Therefore, ∛729 = 9

📝 Example 2: Find ∛1728

Step 1: Prime Factorization using Short Division Method

2 | 1728
2 | 864
2 | 432
2 | 216
2 | 108
2 | 54
3 | 27
3 | 9
3 | 3
  | 1

Step 2: 1728 = (2 × 2 × 2) × (2 × 2 × 2) × (3 × 3 × 3) = 2³ × 2³ × 3³

Step 3: Grouping the factors into triplets, we get (2 × 2 × 3)³ = 12³

Step 4: Therefore, ∛1728 = 12

🔍 Interactive Prime Factorization Tool

Enter a number to see its short division and find the cube root!

👥 Group Work - We Do (16 Minutes)

Collaborative Learning Activities

🎯 Activity 1: Perfect Cubes from 1 to 20

Instructions: Work in groups. Click on a cube to see its prime factorization using the short division method.

1³=1

2³=8

3³=27

4³=64

5³=125

6³=216

7³=343

8³=512

9³=729

10³=1000

11³=1331

12³=1728

13³=2197

14³=2744

15³=3375

16³=4096

17³=4913

18³=5832

19³=6859

20³=8000

Click a number above!

🎯 Activity 2: Larger Cube Numbers

Challenge Problems:

Find ∛35937

Solution:

3 | 35937
3 | 11979
3 | 3993
11| 1331
11| 121
11| 11
  | 1

35937 = (3 × 3 × 3) × (11 × 11 × 11) = 3³ × 11³ = (3 × 11)³
Therefore, ∛35937 = 33

Find ∛42875

Solution:

5 | 42875
5 | 8575
5 | 1715
7 | 343
7 | 49
7 | 7
  | 1

42875 = (5 × 5 × 5) × (7 × 7 × 7) = 5³ × 7³ = (5 × 7)³
Therefore, ∛42875 = 35

Find ∛13824

Solution:

2 | 13824
2 | 6912
2 | 3456
2 | 1728
2 | 864
2 | 432
2 | 216
2 | 108
2 | 54
3 | 27
3 | 9
3 | 3
  | 1

13824 = 2⁹ × 3³ = (2³ × 3)³ = (8 × 3)³ = 24³
Therefore, ∛13824 = 24

✍️ Independent Work - You Do (8 Minutes)

Practice Problems

📋 Solve These Problems:

1. Find the cube root of 512

Solution using Prime Factorization:

2 | 512
2 | 256
2 | 128
2 | 64
2 | 32
2 | 16
2 | 8
2 | 4
2 | 2
  | 1

512 = 2⁹ = (2³)³ = 8³
Therefore, ∛512 = 8

2. Determine the cube root of 729

Solution using Prime Factorization:

3 | 729
3 | 243
3 | 81
3 | 27
3 | 9
3 | 3
  | 1

729 = 3⁶ = (3²)³ = 9³
Therefore, ∛729 = 9

3. Calculate the cube root of 216

Solution using Prime Factorization:

2 | 216
2 | 108
2 | 54
3 | 27
3 | 9
3 | 3
  | 1

216 = 2³ × 3³ = (2 × 3)³ = 6³
Therefore, ∛216 = 6

4. Find the cube root of 1000

Solution using Prime Factorization:

2 | 1000
2 | 500
2 | 250
5 | 125
5 | 25
5 | 5
  | 1

1000 = 2³ × 5³ = (2 × 5)³ = 10³
Therefore, ∛1000 = 10

5. Determine the cube root of 1331

Solution using Prime Factorization:

11 | 1331
11 | 121
11 | 11
   | 1

1331 = 11 × 11 × 11 = 11³
Therefore, ∛1331 = 11

📝 Assessment Questions (9 Minutes)

Factual Questions:

1. What is the cube root of 64?

Answer: ∛64 = 4
Short Division:

2 | 64
2 | 32
2 | 16
2 | 8
2 | 4
2 | 2
  | 1

Prime Factorization: 64 = 2⁶ = (2²)³ = 4³

2. Calculate the cube root of 729 using prime factorization method

Solution:
Short Division:

3 | 729
3 | 243
3 | 81
3 | 27
3 | 9
3 | 3
  | 1

Step 1: Prime factorization: 729 = 3⁶
Step 2: Group in threes: (3²)³ = 9³
Step 3: Therefore, ∛729 = 9

3. Find the cube root of 8000 using prime factorization

Process:
Short Division:

2 | 8000
2 | 4000
2 | 2000
2 | 1000
2 | 500
2 | 250
5 | 125
5 | 25
5 | 5
  | 1

Step 1: Find prime factorization: 8000 = 2⁶ × 5³
Step 2: Group factors in threes: (2² × 5)³ = (4 x 5)³ = 20³
Step 3: Extract cube root: ∛8000 = 20

Open-ended Questions:

1. How can prime factorization help determine if a number is a perfect cube?

Answer: A number is a perfect cube if, in its prime factorization, all the exponents of the prime factors are multiples of 3.
Example: 216 = 2³ × 3³. Both exponents (3 and 3) are multiples of 3, so it's a perfect cube.
Counter-example: 108 = 2² × 3³. The exponent 2 is not a multiple of 3, so it's not a perfect cube.

📚 Textbook Exercise 6.2

Homework Assignment:

• Complete Exercise 6.2 problems 1-2 from your textbook
• Practice the "Think, Discuss and Write" section on page 230
• Review cube roots of numbers 1-20 for next class