Cubes and Cube Roots — Study Notes

1. Introduction

Just as we can square a number by multiplying it by itself, we can cube a number by multiplying it by itself three times. If a number is multiplied by itself three times, the product obtained is called the cube of that number. For example, 2 x 2 x 2 = 8, so 8 is the cube of 2, written as 2^3. Cubes are used everywhere in real life: to find the volume of a cube-shaped box, a tank, or a building block, we multiply length, breadth and height, which are all equal for a cube.

In this chapter we will study cube numbers and their patterns, learn about perfect cubes, find cubes of numbers using properties, and learn how to find cube roots using the method of prime factorisation and estimation. Cubes and cube roots are the natural extension of squares and square roots, and they appear frequently in mensuration, algebra and higher mathematics.

2. Cubes and Perfect Cubes

The cube of a natural number n is written as n^3 = n x n x n. For example, 1, 8, 27, 64, 125, 216, ... are the cubes of 1, 2, 3, 4, 5, 6, ... respectively.

A number which is the cube of a natural number is called a perfect cube. For example, 8, 27, 64 are perfect cubes.

Important Observations about Cubes

1. The cube of an even number is always even, and the cube of an odd number is always odd.
2. The cube of a negative number is always negative. For example, (-2)^3 = -8.
3. If a number ends in 0, 1, 4, 5, 6 or 9, its cube also ends in the same digit. The cube of a number ending in 2 ends in 8, ending in 3 ends in 7, ending in 7 ends in 3, and ending in 8 ends in 2.
4. Cubes are larger than squares for numbers greater than 1, and for proper fractions, the cube is smaller than the number itself.

3. Patterns in Cubes

Interesting Patterns

• The sum of the cubes of the first n natural numbers equals the square of the sum of the first n natural numbers. For example, 1^3 + 2^3 = 1 + 8 = 9 = 3^2.
• Consecutive odd numbers can be grouped to form cubes. For example, 1 = 1^3, 3 + 5 = 8 = 2^3, 7 + 9 + 11 = 27 = 3^3.

These patterns help us understand cubes better and provide quick checks for computations.

4. Cube Roots

The cube root of a number x is that number which when multiplied by itself three times gives x. The cube root is denoted by the symbol cuberoot(x), called the cube root sign. For example, cuberoot(8) = 2 because 2 x 2 x 2 = 8.

Finding Cube Root by Prime Factorisation

To find the cube root of a perfect cube by prime factorisation: 1. Express the number as the product of its prime factors. 2. Group the factors into triples of equal prime numbers. 3. Take one factor from each triple and multiply them.

Example: cuberoot(216) = cuberoot(2 x 2 x 2 x 3 x 3 x 3) = 2 x 3 = 6.

Finding Cube Root by Estimation

For large numbers, the method of estimation is used. We split the number into groups of three digits from the right, and use the cubes of digits 1 to 9 to estimate the answer. For example, to find cuberoot(4913), we note that 4913 lies between 1000 (10^3) and 8000 (20^3), and the last digit 3 tells us the units digit of the root is 7, giving 17.

5. Cube Roots of Negative Numbers and Fractions

Cube Root of a Negative Number

The cube root of a negative number is negative: cuberoot(-64) = -4 because (-4)^3 = -64. Unlike square roots, cube roots of negative numbers do exist.

Cube Root of a Fraction

The cube root of a fraction is the cube root of the numerator divided by the cube root of the denominator. For example, cuberoot(8/27) = cuberoot(8)/cuberoot(27) = 2/3.

Quick Revision Tables

Table 1: Cubes of Numbers 1 to 12

Table 2: Cube Roots of Common Perfect Cubes

Mind Map

graph TD
    A["Cubes and Cube Roots"] --> B["Cube: n x n x n = n^3"]
    A --> C["Perfect cube"]
    C --> D["Cube of even is even, of odd is odd"]
    C --> E["Cube of negative is negative"]
    A --> F["Cube root: cuberoot(x)"]
    F --> G["Prime factorisation: group in triples"]
    F --> H["Estimation method for large numbers"]
    F --> I["Cube root of fractions and negatives"]

Important Diagrams (SVG)

Diagram 1: Visualising Cubes as Volumes

Diagram 2: Cube Root by Prime Factorisation

Common Mistakes

1. Students confuse the cube root with the square root and group prime factors in pairs instead of triples. Cube root needs groups of three.
2. Students forget that the cube of a negative number is negative. For example, cuberoot(-27) = -3, not +3.
3. While using the estimation method, students make mistakes in reading the units digit rule. For example, a number ending in 8 has a cube root ending in 2.
4. Students think that every number has a cube root that is a whole number. Only perfect cubes have whole-number cube roots.
5. Students forget that (a x b)^3 = a^3 x b^3, and try to cube numbers like 2 x 10 incorrectly as 20^3 = 8000 (which is correct) but as 2^3 x 10^3.
6. When finding the cube root of a fraction, students forget to apply the cube root to both numerator and denominator separately.
7. Students do not memorise the cubes and cube roots table and waste time in the exam hall calculating them from scratch.

Exam Tips

1. Memorise cubes of 1 to 15 and cube roots of common perfect cubes like 8, 27, 64, 125, 216, 343, 512, 729 and 1000.
2. In prime factorisation questions, always show the factor tree and the grouping of triples clearly for full method marks.
3. For estimation questions, remember the units-digit correspondence: 1,4,5,6,9,0 stay the same; 2 and 8 swap; 3 and 7 swap.
4. Verify cube root answers by cubing: cuberoot(x) = y means y^3 = x.
5. Read carefully whether the question asks for the cube or the cube root, and whether the number is negative.
6. Practise cube root of decimals and fractions, as these appear in mensuration questions too.
7. Use the property cuberoot(ab) = cuberoot(a) x cuberoot(b) to simplify computations.

Conclusion

Cubes and cube roots complete the study of powers started with squares. In this chapter we learnt how to find the cube of a number and identified the properties of perfect cubes, including the sign rules for negative numbers and the last-digit patterns. We studied how to find cube roots using prime factorisation, where prime factors are grouped in triples, and the estimation method for large numbers. We also dealt with cube roots of negative numbers and fractions. Cubes and cube roots appear again in mensuration for finding volumes, and in algebra for identities and higher-degree equations. A solid command of this chapter will make students confident and fast in examinations.