Squares and Cubes

Squares and cubes are the building blocks of arithmetic and algebra. Understanding the square of a number means multiplying it by itself , while the cube of a number means multiplying it by itself twice more (three times in total). These concepts are essential for solving real-life math problems and are especially helpful for students preparing for competitive exams such as Olympiads, NTSE, JEE, and SSC.

Let’s understand how to calculate squares and cubes of numbers, explore formulas, see solved examples, and download square and cube charts.

 

Table of Contents

• What Are Squares and Cubes?
• Square of a Number
• Cube of a Number
• Squares and Cubes Chart (1 to 50)
• Solved Examples
• Practice Questions
• Real-Life Applications
• Conclusion
• FAQs on Squares and Cubes

 

What Are Squares and Cubes?

Square:
When we square a number, we multiply it by itself one time. It’s like making a perfect square shape, where the length and the breadth are the same.
 Example: 4² means 4 × 4 = 16.

Cube:
When we cube a number, we multiply it by itself two more times. This is like making a perfect cube box, where length, breadth, and height are all the same.
 Example: 5³ means 5 × 5 × 5 = 125.

The notation used is:

• Square: n²

• Cube: n³

 

Square of a Number

The square of a number is written using the exponent 2.
Example:

• 7² = 7 × 7 = 49

• 12² = 12 × 12 = 144

These are also called perfect square numbers.

 

Cube of a Number

The cube of a number is written using the exponent 3.
Example:

• 3³ = 3 × 3 × 3 = 27

• 6³ = 6 × 6 × 6 = 216

These are known as perfect cube numbers.

 

Squares and Cubes Chart (1 to 10)

Number	Square (n²)	Cube (n³)
1	1	1
2	4	8
3	9	27
4	16	64
5	25	125
6	36	216
7	49	343
8	64	512
9	81	729
10	100	1000

 

Solved Examples

Example 1:
Evaluate: 8³ – 12²

Solution:
Cube of 8 = 8³ = 8 × 8 × 8 = 512
Square of 12 = 12² = 12 × 12 = 144
Now, 8³ – 12² = 512 – 144 = 368

Example 2:
If y² = 1936, find the value of y.

Solution:
Given, y² = 1936
As we know, the square of 44 is 1936.
That means 44² = 44 × 44 = 1936
Therefore, y = 44

Example 3:
A cube has an edge of 10 units. Find the volume of the cube.

Solution:
Given,
Edge of a cube = a = 10 units
Volume of the cube = a³
= 10³
= 10 × 10 × 10
= 1000 cubic units

Therefore, the volume of the cube is 1000 cubic units.

Example 4:
Evaluate: 11² + 6³

Solution:
Square of 11 = 11² = 121
Cube of 6 = 6³ = 216
Now, 11² + 6³ = 121 + 216 = 337

Example 5:
The side of a square garden is 25 m. Find its area.

Solution:
Given,
Side of the square = 25 m
Area of square = side²
= 25²
= 25 × 25
= 625 m²

Therefore, the area of the garden is 625 square meters.

 

Practice Questions

1. Find the area of a square field whose side = 23 m.

2. Evaluate: 12² + 13³

3. What is the sum of cubes of first 5 natural numbers?

4. Is 64 a square and a cube number?

5. What is the cube of 40?

 

Real-Life Applications

• Area and Volume : Used in calculating area (square) and volume (cube)

• Mathematics : Found in geometry, algebra, and measurements

• Engineering : Common in engineering and construction estimations

• Exams : Helps in competitive exams (like NTSE, Olympiads, etc.)

• Science : Applies in physics for energy and work calculations

 

Conclusion

Knowing the squares and cubes of numbers helps you solve problems faster and strengthens your number sense. Whether you're calculating areas, volumes, or simplifying algebraic expressions, square and cube values are vital tools. Learn them, practice daily, and keep a chart nearby for reference.

 

 

Frequently Asked Questions on Squares and Cubes

1. What are squares and cubes?

Answer: Squares are numbers multiplied by themselves once, and cubes are multiplied by themselves thrice.

 

2. Are there numbers that are both perfect squares and cubes?

Answer: Yes, 64 is both a square (8²) and a cube (4³).

 

3. What is the square and cube of 20?

Answer: Square of 20 = 400; Cube of 20 = 8000.

 

4. Is 8 a cube number?

Answer: Yes, 8 = 2³, so it is a perfect cube number.

 

5. How can I memorize square and cube numbers?

Answer: Practice daily using a square and cube chart, and solve real-life math problems using them.

 

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