Grade 8
  1. Number Systems  1.1. Rational Numbers  1.2. Irrational Numbers  1.3. Real Numbers  1.4. Properties of Real Numbers  1.4.1. Commutative Property  1.4.2. Associative Property  1.4.3. Distributive Property  1.5. Representation on the Number Line  1.6. Operations on Real Numbers  1.6.1. Addition and Subtraction  1.6.2. Multiplication and Division  1.7. Rationalization  1.8. Exponents and Powers  1.9. Square Roots and Cube Roots
  2. Algebra  2.1. Algebraic Expressions  2.2. Identities and Simplification  2.2.1. Multiplying Binomials  2.2.2. Using Algebraic Identities  2.3. Factorization  2.3.1. Common Factors  2.3.2. Factorization by Grouping  2.3.3. Factorization of Trinomials  2.3.4. Special Products  2.4. Linear Equations in One Variable  2.4.1. Solving Linear Equations  2.4.2. Word Problems on Linear Equations
  3. Geometry  3.1. Understanding Quadrilaterals  3.1.1. Properties of Quadrilaterals  3.1.2. Types of Quadrilaterals  3.1.2.1. Parallelogram  3.1.2.2. Rectangle  3.1.2.3. Square  3.1.2.4. Rhombus  3.1.2.5. Trapezium  3.2. Constructions  3.2.1. Constructing Quadrilaterals  3.2.2. Constructing Bisectors  3.2.3. Constructing Angles  3.2.4. Constructing Triangles  3.3. Symmetry and Transformations  3.3.1. Reflection  3.3.2. Rotation  3.3.3. Translation  3.3.4. Symmetry in Patterns
  4. Mensuration  4.1. Area and Perimeter  4.1.1. Perimeter of Polygons  4.1.2. Area of Triangles  4.1.3. Area of Quadrilaterals  4.1.4. Area of Circles  4.2. Surface Area and Volume  4.2.1. Cubes  4.2.2. Cuboids  4.2.3. Cylinders  4.2.4. Cones  4.2.5. Spheres
  5. Data Handling  5.1. Organizing Data  5.2. Frequency Distribution Tables  5.3. Graphical Representation of Data  5.3.1. Bar Graphs  5.3.2. Histograms  5.3.3. Pie Charts  5.3.4. Line Graphs (added)  5.4. Probability  5.4.1. Introduction to Probability  5.4.2. Experimental Probability
  6. Coordinate Geometry  6.1. Cartesian Plane  6.2. Plotting Points  6.3. Coordinate System  6.4. Distance Between Points  6.5. Applications of Coordinate Geometry
  7. Introduction to Graphs  7.1. Linear Graphs  7.2. Applications of Graphs  7.3. Distance-Time Graphs  7.4. Speed-Time Graphs
  8. Comparing Quantities  8.1. Ratio and Proportion  8.2. Percentage  8.2.1. Increase and Decrease Percent  8.2.2. Discount in Percentage  8.2.3. Profit and Loss in Percentage  8.2.4. Simple Interest  8.2.5. Compound Interest
  9. Exponents and Powers  9.1. Laws of Exponents  9.2. Negative Exponents  9.3. Standard Form  9.4. Applications of Exponents
  10. Introduction to Squares and Square Roots  10.1. Properties of Square Numbers  10.2. Finding Square Roots  10.2.1. Prime Factorization Method  10.2.2. Division Method  10.3. Estimating Square Roots
  11. Introduction to Cubes and Cube Roots  11.1. Properties of Cube Numbers  11.2. Finding Cube Roots  11.2.1. Prime Factorization Method in Finding Cube Roots

Grade 8 ↓


Introduction to Squares and Square Roots


Understanding the concept of squares

When you multiply a number by itself, you get its square. The square of a number is represented by multiplying that number by itself. For example, to square the number 4, you would calculate:

4 × 4 = 16

The result, 16, is called the square of 4. In mathematical notation, the square of a number a is written as a2. Here are some examples to help you understand squares better:

  • 22 = 2 × 2 = 4
  • 52 = 5 × 5 = 25
  • 102 = 10 × 10 = 100

You can see how the squares get large very quickly. Squaring is a common operation in many areas of mathematics, and understanding it is vital to understanding more complex concepts.

Drawing of squares with examples

Consider the geometric representation of squares. A square number can also represent the area of a square shape, where the side length is the base number. It looks like this:

4

In the diagram above, the length of one side of the square is 4, so the area is 42 = 16. This area is represented by the total number of small squares within the large square, each of which has an area of 1 square unit.

General properties of squares

  • The square is always positive, because multiplying two positive numbers or two negative numbers gives a positive product.
  • When a whole number is squared, the result is called a perfect square.
  • The sequence of perfect squares is: 1, 4, 9, 16, 25, 36, ...

Exploring the concept of square root

The square root of a number is the value that, when multiplied by itself, gives the original number. If a2 = b, then a is the square root of b. The symbol for a square root is √. For example:

√16 = 4

This means that 4 × 4 = 16.

Illustrating square roots with examples

4

This figure shows the square root of 16, which is 4. Each side of the square above is 4 units long. Thus, its area is 16 square units, which shows that the square root of 16 is equal to 4.

Calculating the square root

Calculating the square root for non-perfect squares can be more complicated and often involves estimating or using a calculator. For example, the square root of 20 is approximately 4.472.

Here's how to manually estimate the square root for an imperfect square, such as 20:

  1. Find two consecutive whole numbers between which the square root lies. For √20, the two numbers are 4 and 5 because 42 = 16 and 52 = 25.
  2. Estimate the midpoint or average of these two numbers, say 4.5, and square it: 4.52 = 20.25.
  3. Since 20.25 is close to 20, 4.5 is a good estimate, but we can refine it further with more calculations.

Use of square roots in problem solving

Square roots are often used in various mathematical calculations, including finding distances, solving quadratic equations, and more. Here's a practical example:

Suppose you have a square with an area of 64 square units, and you want to find the length of one side. Since the formula for area is side times side, you need to find the square root of the area:

√64 = 8

The length of one side of the square is 8 units.

Practical examples and exercises

Let's do some exercises. Try them out yourself:

  1. What is the square of 7?
  2. Find the square root of 49.
  3. If the area of a square is 144 square units, what is the length of each side?
  4. Approximate the square root of 30 to one decimal place.

Conclusion

Squares and square roots are fundamental concepts that you will use in your study of mathematics and in real-world applications. Understanding these concepts helps develop important problem-solving skills. Keep practicing, and use the operations of finding squares and square roots to explore interesting mathematical challenges.


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Introduction to Squares and Square Roots

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