Class 8 Math Squares and Square Roots Notes
Squares and Square Roots – Class 8
Hi everyone! This chapter is about Squares and Square Roots. These are important math concepts that are used in many real-life situations.
What are Squares?
The square of a number is the result when you multiply the number by itself. We write it with a small ‘2’ as a superscript (like this: x²).
Examples: 3² = 3 * 3 = 9, 5² = 5 * 5 = 25, 10² = 10 * 10 = 100
A perfect square is a number that can be obtained by squaring an integer. For example, 9, 25, and 100 are perfect squares.
What are Square Roots?
The square root of a number is a value that, when multiplied by itself, gives the original number. We write it with a radical symbol (√).
Examples: √9 = 3 (because 3 * 3 = 9), √25 = 5 (because 5 * 5 = 25), √100 = 10 (because 10*10 = 100)
Finding the square root is the opposite of squaring a number.
Properties of Squares and Square Roots
- The square of an even number is always even.
- The square of an odd number is always odd.
- The square root of a perfect square is always an integer.
- The square root of a number can be positive or negative (but we usually consider the positive root in most practical applications).
Finding Square Roots
1. Prime Factorization Method:
Break down the number into its prime factors. Then, group the factors in pairs. For each pair, take one factor out. Multiply the factors you took out to get the square root.
Example: Find √144
- Prime factors of 144: 2 * 2 * 2 * 2 * 3 * 3
- Pairs: (2 * 2) * (2 * 2) * (3 * 3)
- Square root: 2 * 2 * 3 = 12
2. Long Division Method:
This is a method you can use for larger numbers, but it’s a bit more complicated. You can search online for tutorials on how to do long division for square roots if you need to learn this method.
Applications of Squares and Square Roots
1. Geometry:
Finding the area of a square, the length of the diagonal of a square or rectangle (using the Pythagorean theorem), and other geometric calculations.
2. Construction and Design:
Calculating dimensions, areas, and material requirements.
3. Physics:
Many physics formulas involve squares and square roots (e.g., calculating kinetic energy).
4. Computer Graphics:
Used in calculations related to distance, size, and scaling of objects.
Squares and square roots are fundamental concepts that you’ll use throughout your math studies and in many real-world situations.
Squares and Square Roots Quiz – Application Problems
1. Tile Floor: A square tile floor is made up of 144 individual square tiles. How many tiles are along each side of the floor?
2. Garden Plot: A square garden plot has an area of 625 square meters. What is the length of each side of the garden?
3. Right Triangle: A right triangle has legs of length 3 cm and 4 cm. What is the length of the hypotenuse (the side opposite the right angle)?
4. Square Picture: A square picture frame has an area of 400 square centimeters. What is the length of one side of the frame?
5. Construction: A builder needs to calculate the length of the diagonal of a square foundation that is 10 meters on each side. What is the length of the diagonal?
6. Area Calculation: A square room has a side length of 8 meters. What is the area of the room?
7. Pythagorean Theorem: A ladder leans against a wall. The ladder is 13 feet long, and the base of the ladder is 5 feet away from the wall. How high up the wall does the ladder reach?
8. Scaling: A square image is scaled up so that its side length doubles. How many times larger does the area of the image become?
9. Tiling: A bathroom floor is a square with a side length of 3 meters. Square tiles with a side length of 25 cm are used to cover the floor. How many tiles are needed?
10. Diagonal of a Square: A square has a side length of 7 cm. What is the length of its diagonal?
