CUBE AND CUBE ROOTS

Cube of a Number

• A cube of a number is the result of multiplying the number by itself three times.
• If 'a' is a number, its cube is \(a \times a \times a = a^3\).
• Geometric Interpretation: The volume of a cube with side length 'a' is \(a^3\).

Examples of Cubes

• \(1^3 = 1 \times 1 \times 1 = 1\)
• \(2^3 = 2 \times 2 \times 2 = 8\)
• \(3^3 = 3 \times 3 \times 3 = 27\)
• \(4^3 = 4 \times 4 \times 4 = 64\)
• \(5^3 = 5 \times 5 \times 5 = 125\)
• \(10^3 = 10 \times 10 \times 10 = 1000\)

Important Observations

• Cube of an even number is always an even number. (e.g., \(2^3=8\), \(4^3=64\))
• Cube of an odd number is always an odd number. (e.g., \(3^3=27\), \(5^3=125\))
• Cube of a negative number is always a negative number. (e.g., \((-2)^3 = -8\))
• Cube of a rational number \(\frac{p}{q}\) is \((\frac{p}{q})^3 = \frac{p^3}{q^3}\).
• Cube of a number ending in 0 will end in three zeros. (e.g., \(10^3 = 1000\), \(20^3 = 8000\))
• Cube of a number ending in 1 ends in 1. (e.g., \(1^3=1\), \(11^3=1331\))
• Cube of a number ending in 2 ends in 8. (e.g., \(2^3=8\), \(12^3=1728\))
• Cube of a number ending in 3 ends in 7. (e.g., \(3^3=27\), \(13^3=2197\))
• Cube of a number ending in 4 ends in 4. (e.g., \(4^3=64\), \(14^3=2744\))
• Cube of a number ending in 5 ends in 5. (e.g., \(5^3=125\), \(15^3=3375\))
• Cube of a number ending in 6 ends in 6. (e.g., \(6^3=216\), \(16^3=4096\))
• Cube of a number ending in 7 ends in 3. (e.g., \(7^3=343\), \(17^3=4913\))
• Cube of a number ending in 8 ends in 2. (e.g., \(8^3=512\), \(18^3=5832\))
• Cube of a number ending in 9 ends in 9. (e.g., \(9^3=729\), \(19^3=6859\))

• Pattern of last digits: Notice that 0, 1, 4, 5, 6, 9 remain the same. 2 changes to 8, 8 to 2. 3 changes to 7, 7 to 3. This is useful for estimating cube roots.

What is a Perfect Cube?

• A natural number is called a perfect cube (or a cube number) if it is the cube of some natural number.
• For example, 8 is a perfect cube because \(8 = 2^3\). 27 is a perfect cube because \(27 = 3^3\).
• Numbers like 2, 3, 4, 5, 6, 7, 9, 10, etc., are not perfect cubes.

How to Check if a Number is a Perfect Cube (Prime Factorization Method)

1. Find the prime factorization of the given number.
2. Group the prime factors in triplets (groups of three identical factors).
3. If all prime factors can be grouped into triplets, then the number is a perfect cube.
4. If any prime factor is left ungrouped (i.e., it does not appear in a group of three), then the number is not a perfect cube.

Making a Number a Perfect Cube

• To multiply: If a number is not a perfect cube, find its prime factors. For each factor that is not part of a triplet, multiply the number by the missing factors to complete the triplet(s).
• Example: \(72 = 2 \times 2 \times 2 \times 3 \times 3\). Here, 3 appears twice. To make it a triplet, we need one more 3. So, multiply 72 by 3: \(72 \times 3 = 216 = 6^3\).
• To divide: If a number is not a perfect cube, find its prime factors. For each factor that is in excess (i.e., not forming a complete triplet, or forming more than one triplet where one is incomplete), divide the number by the excess factors.
• Example: \(250 = 2 \times 5 \times 5 \times 5\). Here, 2 is not part of a triplet. To make it a perfect cube, divide 250 by 2: \(250 \div 2 = 125 = 5^3\).

Interesting Patterns and Properties

• Sum of consecutive odd numbers: The cube of a natural number 'n' can be expressed as the sum of 'n' consecutive odd numbers.
• \(1^3 = 1\)
• \(2^3 = 8 = 3 + 5\)
• \(3^3 = 27 = 7 + 9 + 11\)
• \(4^3 = 64 = 13 + 15 + 17 + 19\)
• The first odd number in the sum for \(n^3\) is given by \(n(n-1) + 1\).
• Cubes of numbers ending in 0, 1, 4, 5, 6, 9: Their cubes also end in the same digit.
• Cubes of numbers ending in 2: Their cubes end in 8.
• Cubes of numbers ending in 8: Their cubes end in 2.
• Cubes of numbers ending in 3: Their cubes end in 7.
• Cubes of numbers ending in 7: Their cubes end in 3.

Hardy-Ramanujan Number

• The number 1729 is known as the Hardy-Ramanujan number.
• It is the smallest number that can be expressed as the sum of two cubes in two different ways:
• \(1729 = 1^3 + 12^3 = 1 + 1728\)
• \(1729 = 9^3 + 10^3 = 729 + 1000\)
• Any number that can be expressed as the sum of two cubes in two different ways is called a taxi-cab number.

What is a Cube Root?

• The cube root of a number 'x' is the number 'y' such that \(y^3 = x\).
• It is denoted by the symbol \(\sqrt[3]{}\) (read as 'cube root of').
• For example, \(\sqrt[3]{8} = 2\) because \(2^3 = 8\).
• \(\sqrt[3]{27} = 3\) because \(3^3 = 27\).

Properties of Cube Roots

• Cube root of a negative number: The cube root of a negative number is always a negative number.
• \(\sqrt[3]{-8} = -2\) because \((-2)^3 = -8\).
• Cube root of a product: \(\sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b}\).
• Cube root of a quotient: \(\sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}}\).
• Cube root of a decimal number: Convert the decimal to a fraction, find cube root of numerator and denominator, then convert back to decimal.
• Example: \(\sqrt[3]{0.008} = \sqrt[3]{\frac{8}{1000}} = \frac{\sqrt[3]{8}}{\sqrt[3]{1000}} = \frac{2}{10} = 0.2\).

Estimating Cube Roots (Mental Math Technique)

This method is useful for perfect cubes of larger numbers.

1. Look at the last digit of the number: This tells you the last digit of its cube root.

• Ends in 1 \(\rightarrow\) root ends in 1
• Ends in 8 \(\rightarrow\) root ends in 2
• Ends in 7 \(\rightarrow\) root ends in 3
• Ends in 4 \(\rightarrow\) root ends in 4
• Ends in 5 \(\rightarrow\) root ends in 5
• Ends in 6 \(\rightarrow\) root ends in 6
• Ends in 3 \(\rightarrow\) root ends in 7
• Ends in 2 \(\rightarrow\) root ends in 8
• Ends in 9 \(\rightarrow\) root ends in 9
• Ends in 0 \(\rightarrow\) root ends in 0

1. Strike off the last three digits of the number.
2. Consider the remaining number: Find the largest perfect cube less than or equal to this remaining number. The cube root of this perfect cube will be the tens digit of your answer.

• Example: Find \(\sqrt[3]{17576}\)
• Last digit is 6, so cube root ends in 6.
• Strike off last three digits (756), remaining number is 17.
• Largest perfect cube \(\le 17\) is 8 (which is \(2^3\)). So, the tens digit is 2.
• Therefore, \(\sqrt[3]{17576} = 26\).

Step-by-Step Method

This is the most reliable method for finding the cube root of any perfect cube.

1. Prime Factorize the Number: Find all the prime factors of the given number.
2. Group Factors in Triplets: Group identical prime factors in sets of three.
3. Take One Factor from Each Triplet: For each group of three identical factors, take out one factor.
4. Multiply the Taken Factors: Multiply all the factors taken out from the triplets. This product is the cube root of the original number.

Example: Find \(\sqrt[3]{216}\)

1. Prime Factorization: \(216 = 2 \times 108 = 2 \times 2 \times 54 = 2 \times 2 \times 2 \times 27 = 2 \times 2 \times 2 \times 3 \times 9 = 2 \times 2 \times 2 \times 3 \times 3 \times 3\).
2. Group in Triplets: \(216 = (2 \times 2 \times 2) \times (3 \times 3 \times 3)\).
3. Take one factor: From the first triplet of 2s, take 2. From the second triplet of 3s, take 3.
4. Multiply: \(2 \times 3 = 6\).

Therefore, \(\sqrt[3]{216} = 6\).

Cube Root of a Product of Numbers

• If 'a' and 'b' are two numbers, then \(\sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b}\).

Cube Root of a Rational Number

• If \(\frac{p}{q}\) is a rational number, then \(\sqrt[3]{\frac{p}{q}} = \frac{\sqrt[3]{p}}{\sqrt[3]{q}}\).

Cube Root of a Decimal

• Convert the decimal to a fraction, find the cube root of the numerator and denominator, then convert back to decimal.
• Example: \(\sqrt[3]{0.125} = \sqrt[3]{\frac{125}{1000}} = \frac{\sqrt[3]{125}}{\sqrt[3]{1000}} = \frac{5}{10} = 0.5\).