Grade 7
  1. Number System  1.1. Integers  1.1.1. Properties of Integers  1.1.2. Operations on Integers  1.1.3. Additive and Multiplicative Inverses  1.1.4. Applications of Integers  1.2. Rational Numbers  1.2.1. Definition of Rational Numbers  1.2.2. Properties of Rational Numbers  1.2.3. Operations on Rational Numbers  1.2.4. Standard Form of Rational Numbers  1.3. Decimals  1.3.1. Operations on Decimals  1.3.2. Converting Between Fractions and Decimals  1.4. Powers and Exponents  1.4.1. Laws of Exponents  1.4.2. Simplifying Expressions with Exponents  1.4.3. Scientific Notation  1.5. Roots  1.5.1. Square Roots  1.5.2. Cube Roots
  2. Algebra  2.1. Expressions  2.1.1. Algebraic Expressions  2.1.2. Simplifying Expressions  2.1.3. Evaluating Expressions  2.2. Linear Equations  2.2.1. Solving Simple Equations  2.2.2. Applications of Linear Equations  2.3. Algebraic Identities  2.3.1. Expanding Using Identities  2.3.2. Simplifying Using Identities
  3. Ratio and Proportion  3.1. Ratios  3.1.1. Understanding Ratios  3.1.2. Simplifying Ratios  3.1.3. Equivalent Ratios  3.2. Proportion  3.2.1. Concept of Proportion  3.2.2. Direct Proportion  3.2.3. Inverse Proportion  3.3. Unitary Method  3.3.1. Solving Problems Using the Unitary Method
  4. Geometry  4.1. Lines and Angles  4.1.1. Types of Angles  4.1.2. Pairs of Angles  4.1.3. Properties of Parallel Lines and a Transversal  4.1.4. Angle Bisectors  4.2. Triangles  4.2.1. Types of Triangles  4.2.2. Angle Sum Property  4.2.3. Exterior Angle Property  4.2.4. Pythagoras Theorem  4.3. Congruence of Triangles  4.3.1. Criteria for Congruence  4.3.2. Applications of Congruence  4.4. Quadrilaterals  4.4.1. Types of Quadrilaterals  4.4.2. Properties of Parallelograms  4.4.3. Special Quadrilaterals  4.4.4. Properties of Rectangles and Rhombuses
  5. Mensuration  5.1. Perimeter and Area  5.1.1. Perimeter of Plane Figures  5.1.2. Area of Plane Figures  5.1.3. Area of a Trapezium  5.1.4. Area of a Parallelogram  5.1.5. Area of a Circle  5.1.6. Area of Composite Figures  5.2. Surface Area and Volume  5.2.1. Cubes and Cuboids  5.2.2. Surface Area of Cylinders  5.2.3. Volume of Prisms and Cylinders  5.2.4. Volume and Surface Area of Cones
  6. Data Handling  6.1. Data Collection  6.1.1. Organizing Data  6.1.2. Frequency Distribution  6.2. Graphical Representation of Data  6.2.1. Bar Graphs  6.2.2. Double Bar Graphs  6.2.3. Pie Charts  6.2.4. Histograms  6.2.5. Line Graphs  6.3. Probability  6.3.1. Simple Events  6.3.2. Experimental Probability  6.3.3. Theoretical Probability
  7. Practical Geometry  7.1. Construction of Triangles  7.1.1. Constructing Triangles Given Side Lengths  7.1.2. Constructing Triangles Given Side and Angle  7.1.3. Constructing Right-Angled Triangles  7.2. Construction of Quadrilaterals  7.2.1. Quadrilaterals Given Side Lengths and Angles  7.2.2. Constructing Parallelograms and Rectangles

Grade 7 → Number System → Roots ↓


Cube Roots


In the world of mathematics, roots are fascinating and essential concepts. One of the different types of roots is the beloved "cube root". But what is a cube root and how does it work? Let's take a deeper look into this wonderful topic and learn about cube roots in detail.

What is cube root?

The cube root of a number is a value that, when multiplied by itself twice, gives the original number. In simple terms, if x is the cube root of y, then:

x × x × x = y

This can also be written using the radical symbol as:

³√y = x

where the small '3' above the radical sign indicates the cube root.

Illustrating cube roots

Visual representations are one of the best ways to understand mathematical concepts. Let's use a basic cube to visualize cube roots.

Volume= A × A × A

Imagine you have a cube, and its volume (the space inside) can be found by multiplying its side length (a) by itself twice. The cube root helps you find the side length (a) if you know the volume.

Cube root examples

Let's look at some simple examples:

Example 1

Find the cube root of 8.

³√8 = 2

Because 2 × 2 × 2 = 8.

Example 2

Find the cube root of 27.

³√27 = 3

Because 3 × 3 × 3 = 27.

Example 3

Find the cube root of 64.

³√64 = 4

Because 4 × 4 × 4 = 64.

Properties of cube root

1. Cube root of 1

The cube root of 1 is always 1:

³√1 = 1

2. Cube root of 0

The cube root of 0 is always 0:

³√0 = 0

3. Positive and negative roots

If a number is positive, its cube root will be positive. If a number is negative, its cube root will also be negative. This is different from square roots, where negative numbers have no real roots.

For example:

³√(-8) = -2

Because -2 × -2 × -2 = -8.

4. Cube root of an imperfect cube

For non-perfect cubes, the cube roots are not whole numbers and can be expressed as decimals or fractions. Calculators or estimation methods are typically used to find these roots. Consider ³√5, which is approximately 1.71.

Applications of cube root

Understanding cube roots can be helpful in real-life situations, such as:

  • Calculating the side length of a cube when you know its volume.
  • Understanding scientific calculator and advanced mathematics.
  • Design and architecture for space and volume calculations.

Finding cube root without a calculator

While cube roots can be found quickly with a calculator, it is also beneficial to know methods for estimating or calculating them manually for simpler problems.

Method: Prime factorization

Let's find the cube root of 216 using prime factorization.

Step 1: Break down 216 into prime numbers:

216 ÷ 2 = 108 108 ÷ 2 = 54 54 ÷ 2 = 27 27 ÷ 3 = 9 9 ÷ 3 = 3 3 ÷ 3 = 1

The prime factors of 216 are 2 × 2 × 2 × 3 × 3 × 3.

Step 2: Group the factors into triples:

(2 × 2 × 2) and (3 × 3 × 3)

Step 3: Take one factor from each triplet:

2 × 3 = 6

The cube root of 216 is 6.

Estimating the cube root

What if you want to find the cube root of a large number, such as 1000? You can estimate it like this:

  • First, identify the two perfect cubes between which your number lies. In this case, 1000 lies between 729 (9³) and 1331 (11³).
  • Since 1000 is close to 10³ = 1000, the cube root of 1000 is approximately 10.

Practice problems

Let's practice finding cube roots with some easy problems:

  1. Find the cube root of 512.
  2. What is the cube root of 343?
  3. Find the cube root of 50. (Hint: 3³ = 27 and 4³ = 64)

Answer key

  1. ³√512 = 8 because 8 × 8 × 8 = 512.
  2. ³√343 = 7 because 7 × 7 × 7 = 343.
  3. The cube root of 50 is between 3 and 4. Close to 3 because 50 is closer to 27 than it is to 64. You can estimate it to be around 3.68.

Conclusion

Understanding the concept of cube root is essential as it lays the foundation for higher mathematics and applications in various fields. Learning to calculate and estimate cube roots helps strengthen mental math skills and enhance problem-solving abilities. With practice, the world of cube roots reveals the beauty and logic of mathematical calculations.


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Cube Roots

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