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 ↓


Finding Square Roots


Understanding square roots is a fundamental aspect of mathematics that we are going to explore in detail. Square roots form the basis of many concepts in mathematics and can appear in various fields such as geometry, algebra, and calculus. In this guide, we will discuss in depth what square roots are, how to find them, and their significance.

What is a square root?

The square root of a number is the value that when multiplied by itself gives the number. In simple terms, if n is the square root of x, then n times n equals x. This relationship can be expressed by the formula:

n * n = x

For example, the square root of 16 is 4 because when you multiply 4 by itself, you get 16:

4 * 4 = 16

Square roots of perfect squares

Perfect squares are numbers whose square roots are whole numbers. Here is a list of the first few perfect squares and their square roots:

  • 1² = 1, so the square root of 1 is 1.
  • 2² = 4, so the square root of 4 is 2.
  • 3² = 9, so the square root of 9 is 3.
  • 4² = 16, so the square root of 16 is 4.
  • 5² = 25, so the square root of 25 is 5.
  • ...and so on.

Visual example:

4 = 16

In the above example, the area of a square with 4 sides is 16. Thus, 4 is the square root of 16.

Finding square roots of non-perfect squares

When a number is not a perfect square, its square root is not a whole number. The square roots of these numbers are irrational numbers, which means they cannot be expressed as simple fractions and have non-repeating, non-terminating decimal parts.

Finding the square root using estimation:

The simplest way to find the square root of a non-perfect square is to estimate it. It works like this:

  1. Identify the two perfect squares your number lies between.
  2. Estimate the square root as a decimal.
  3. Refine your estimate by checking the square of your estimate and adjusting accordingly.

Example:

Let's find the square root of 20 using estimation:

  • We know that 4² = 16 and 5² = 25, 20 is between 16 and 25.
  • Therefore, the square root of 20 lies between 4 and 5.
  • If we try 4.5 (the midpoint of 4 and 5):
4.5 * 4.5 = 20.25

Since 20.25 is slightly more than 20, we try a slightly smaller number, say 4.4:

4.4 * 4.4 = 19.36

Now 19.36 is less than 20, so the square root of 20 when further refined will be approximately 4.47.

Square roots and the real number line

Understanding square roots also involves recognizing their location on the real number line. Consider the following visual example:

Visual example:

0 1 2 3 4 √2

In the above number line illustration, we can see that √2 is placed between 1 and 2, which is close to 1.4.

Calculating the square root using long division method

The long division method is a systematic way to find the approximate value of the square root of a number. Here is a step-by-step approach:

  1. First group the digits in pairs from right to left, adding decimal points to make odd lengths equal.
  2. Find the largest number whose square is less than or equal to the leftmost digit group.
  3. Bring the next group of digits into the remainder to form the dividend.
  4. Multiply the quotient (ignoring the last digit) by two and keep a temporary digit for the new divisor, so that the result when multiplied by this new digit is less than or equal to the dividend.
  5. Repeat the process for desired precision.

Example:

Let's calculate √625 using the long division method:

  1. Group Marks: (6 25)
  2. 6: The largest number whose square is less than or equal to 6 is 2 (because 2² = 4).
  3. Subtract: 6 - 4 = 2, subtract 25 to make 225.
  4. New divisor: 2 * 2 = 4_ Keep one digit (5), adjusting where necessary: 45 * 5 = 225.
  5. There is no remainder, so the square root of 625 is 25.

Special cases and properties

Some properties and special cases of the square root simplify the calculations:

  • Square root of 0: The square root of 0 is always 0.
  • Square roots of negative numbers: Square roots of negative numbers do not lie in real numbers. They are expressed using imaginary numbers.
  • Product Property: The square root of a product is equal to the product of the square roots.
    √(a * b) = √a * √b
  • Quotient Property: The square root of a quotient is equal to the quotient of the square roots.
    √(a / b) = √a / √b when b ≠ 0

Applications of square roots

Square roots appear in many aspects of real life and academic applications:

  • Geometry: Calculating the area of a square or the side of a rectangle by finding its diagonals.
  • Physics: Nave equations involve square roots, as do the equations of motion.
  • Engineering: Designs involving resistors, capacitors and inductors that use square roots to determine parameters.

Conclusion

Finding square roots is an important part of mathematics. By understanding the estimation method and the long division method, we can find the square roots of both perfect squares and imperfect squares. This knowledge extends to many practical applications, making it an essential skill in both academic and real-life scenarios. By practicing these methods, anyone can gain accuracy and efficiency in finding square roots.


Grade 8 → 10.2


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Finding Square Roots

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