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 → Number Systems ↓


Square Roots and Cube Roots


Understanding square roots and cube roots is an important part of mastering number system concepts in maths. This topic helps us simplify maths problems and solve more complex equations. Let's understand these concepts step by step.

Square root

The square root of a number is the value that when multiplied by itself gives the original number. The symbol for square root is √.

Mathematically, if x is the square root of y, then:

x * x = y

Examples of square roots

Let's start with some simple examples:

  • The square root of 16 is 4 because 4 * 4 = 16.
  • The square root of 25 is 5 because 5 * 5 = 25.
  • The square root of 36 is 6 because 6 * 6 = 36.

These examples can be represented visually as follows:

4X4,16 5X5,25

In these sequences, each square is a number multiplied by itself to form a perfect square.

Finding square roots of non-perfect squares

Finding the square root of numbers that are not perfect squares requires more effort. It often involves guessing or using a calculator. For example, the square root of 20 is approximately 4.47 because 4.47 * 4.47 ≈ 20.

In general, for non-perfect squares, you can list the perfect squares close to the number and use them to estimate. For example:

  • The square root of 20 lies between 4 (because 4 * 4 = 16) and 5 (because 5 * 5 = 25).

Using the square root property

The square root property states that for any non-negative numbers a and b:

a^2 = b ⇒ a = √b

This property helps solve equations that involve squares. For example, in solving x^2 = 49, we use the square root property to find:

x = √49 = 7

Cube root

Cube roots work just like square roots, but involve multiplying a number by itself twice. The cube root of a number is the value that gives the original number when used three times in a multiplication. The symbol for a cube root is ∛.

In mathematical terms, if y is the cube root of z, then:

y * y * y = z

Cube root examples

Here are some simple examples of cube roots:

  • The cube root of 8 is 2 because 2 * 2 * 2 = 8.
  • The cube root of 27 is 3 because 3 * 3 * 3 = 27.
  • The cube root of 64 is 4 because 4 * 4 * 4 = 64.

These examples can be visualized as follows:

2X2X2,8 3X3X3,27

For perfect cubes, each cube can be represented as a value multiplied by itself three times.

Finding cube roots of imperfect cubes

It is difficult to calculate the cube root of non-perfect cubes without a calculator, but you can still use an estimation. Consider the cube root of 50 It is between:

  • 3 (since 3 * 3 * 3 = 27)
  • 4 (since 4 * 4 * 4 = 64)

Using the cube root property

The cube root property is useful in solving equations involving cubic powers. For any number b:

a^3 = b ⇒ a = ∛b

For example, if we want to solve x^3 = 125, we use the following property to find:

x = ∛125 = 5

Assorted examples and exercises

Let's practice with some mixed examples to strengthen our understanding:

  1. Find the square root of 81:
    The square root is 9 because 9 * 9 = 81.
  2. Find the cube root of 216:
    The cube root is 6 because 6 * 6 * 6 = 216.
  3. Find the square root of 121.
    The square root is 11.
  4. Find the cube root of 343.
    The cube root is 7 because 7 * 7 * 7 = 343.

Summary

Square roots and cube roots help us understand the relationships between numbers in a number system. The square root involves finding a number that is multiplied by itself to get the original number. The cube root requires finding a number that is multiplied by itself three times to get the original number. Understanding these processes helps in solving various mathematical problems.

With more practice, the concepts of square roots and cube roots become easier to understand, allowing you to solve more complex mathematical problems with confidence.


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