Cubes And Cube Roots Class 8 Maths Notes

Key Definitions

Cube Number

A cube number (or a perfect cube) is a number obtained by multiplying an integer by itself three times. For example, $2 \times 2 \times 2 = 8$, so 8 is a cube number.

Perfect Cube

A natural number is called a perfect cube if it is the cube of some natural number. For example, 1, 8, 27, 64, 125, etc., are perfect cubes.

Cube Root

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

Prime Factorization Method

A method used to find the cube root of a perfect cube by expressing the number as a product of its prime factors and then grouping them in triplets.

Units Digit

The digit in the one's place of a number. This digit is often helpful in determining properties of cube numbers or their cube roots.

Understanding Cubes and Their Properties

A cube number is the result of multiplying an integer by itself three times. This is represented mathematically as $n^3$, where $n$ is an integer. For instance, $3^3 = 3 \times 3 \times 3 = 27$. The number 27 is a perfect cube. Understanding the properties of these numbers is crucial.

Properties of Cube Numbers:

1. Units Digit Pattern: The units digit of a cube number follows a specific pattern. For example:

• If a number ends with 0, 1, 4, 5, 6, 9, its cube ends with the same digit (e.g., $1^3=1, 4^3=64, 5^3=125$).
• If a number ends with 2, its cube ends with 8 (e.g., $2^3=8$).
• If a number ends with 8, its cube ends with 2 (e.g., $8^3=512$).
• If a number ends with 3, its cube ends with 7 (e.g., $3^3=27$).
• If a number ends with 7, its cube ends with 3 (e.g., $7^3=343$).

This pattern helps in quickly predicting the units digit of a cube or its cube root.

1. Cubes of Even and Odd Numbers: The cube of an even number is always even (e.g., $2^3=8, 4^3=64$), and the cube of an odd number is always odd (e.g., $3^3=27, 5^3=125$). This property is useful for quick checks.

1. Sum of Consecutive Odd Numbers: Interestingly, some perfect cubes can be expressed as the sum of consecutive odd numbers. For example:

• $1^3 = 1$
• $2^3 = 8 = 3 + 5$
• $3^3 = 27 = 7 + 9 + 11$
• $4^3 = 64 = 13 + 15 + 17 + 19$

This pattern can sometimes be used to identify perfect cubes or for mental math.

1. Negative Numbers: The cube of a negative number is always negative (e.g., $(-2)^3 = -8$). This contrasts with squares, where $(-2)^2 = 4$ is positive.

To identify if a number is a perfect cube, one reliable method is prime factorization. If, after prime factorization, all prime factors appear in groups of three (triplets), then the number is a perfect cube. If any prime factor is left without forming a triplet, the number is not a perfect cube. For instance, to check if 216 is a perfect cube, we factorize it: $216 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 = 2^3 \times 3^3 = (2 \times 3)^3 = 6^3$. Since all factors form triplets, 216 is a perfect cube.

Key Points to Remember

• A number $x$ is a perfect cube if it can be written as $n^3$ for some integer $n$.
• The cube of a number ending in 0, 1, 4, 5, 6, 9 also ends in the same digit.
• The cube of a number ending in 2 ends in 8, and vice-versa (8 ends in 2).
• The cube of a number ending in 3 ends in 7, and vice-versa (7 ends in 3).
• The cube of an even number is even; the cube of an odd number is odd.
• To find the cube root of a number by prime factorization, group the prime factors into triplets.
• If a number is a perfect cube, its prime factors can be grouped into triplets.
• The cube root of a negative perfect cube is always negative (e.g., $\sqrt[3]{-27} = -3$).
• The symbol for cube root is $\sqrt[3]{ }$.

Finding Cube Root by Prime Factorization Method

1. — Write the given number as a product of its prime factors.
2. — Group the prime factors into triplets (groups of three identical factors).
3. — From each triplet, take one prime factor.
4. — Multiply these selected prime factors. The product will be the cube root of the given number.

Worked Examples

• {"title":"Example 1: Find the cube of 7.","description":"Solution: To find the cube of 7, we multiply 7 by itself three times.\n$7^3 = 7 \\times 7 \\times 7 = 49 \\times 7 = 343$.\nTherefore, the cube of 7 is 343."}
• {"title":"Example 2: Is 729 a perfect cube? If yes, find its cube root.","description":"Solution:\n1. Prime factorization of 729: $729 = 3 \\times 3 \\times 3 \\times 3 \\times 3 \\times 3$.\n2. Grouping in triplets: $729 = (3 \\times 3 \\times 3) \\times (3 \\times 3 \\times 3) = 3^3 \\times 3^3$.\n3. Since all prime factors form triplets, 729 is a perfect cube.\n4. Taking one factor from each triplet: $3 \\times 3 = 9$.\nTherefore, $\\sqrt[3]{729} = 9$."}
• {"title":"Example 3: Find the smallest number by which 256 must be multiplied to make it a perfect cube.","description":"Solution:\n1. Prime factorization of 256: $256 = 2 \\times 2 \\times 2 \\times 2 \\times 2 \\times 2 \\times 2 \\times 2 = 2^8$.\n2. Grouping in triplets: $256 = (2 \\times 2 \\times 2) \\times (2 \\times 2 \\times 2) \\times 2 \\times 2 = 2^3 \\times 2^3 \\times 2^2$.\n3. Here, two factors of 2 are left ($2^2$). To make it a triplet, we need one more 2.\nTherefore, 256 must be multiplied by 2 to make it a perfect cube ($256 \\times 2 = 512 = 8^3$)."}

Exam Tip: Avoiding Common Mistakes

When solving problems related to cubes and cube roots, especially using the prime factorization method, be meticulous with your calculations. A single error in prime factorization can lead to an incorrect answer. Always double-check your factorization steps. Remember that $\sqrt[3]{x}$ is not the same as $x/3$. For multiple-choice questions, use the units digit pattern as a quick elimination strategy. For instance, if the question asks for the cube root of a number ending in 7, the answer must end in 3. Practice identifying perfect cubes quickly by observing their prime factors; this saves time in exams.

Practice Questions with Solutions

• Q: What is the units digit of the cube of 13? A: The number 13 ends with 3. The cube of a number ending in 3 ends with 7. So, the units digit of $13^3$ is 7.
• Q: Is 500 a perfect cube? Justify your answer. A: To check if 500 is a perfect cube, we find its prime factors: $500 = 2 \times 2 \times 5 \times 5 \times 5 = 2^2 \times 5^3$. Since the prime factor 2 does not form a triplet, 500 is not a perfect cube.
• Q: Find the cube root of 1728 using prime factorization. A: Prime factorization of 1728: $1728 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 = 2^3 \times 2^3 \times 3^3$. Grouping in triplets gives $(2 \times 2 \times 3)^3 = 12^3$. So, $\sqrt[3]{1728} = 12$.
• Q: The cube of an odd number is always an __________ number. A: The cube of an odd number is always an odd number.

Frequently Asked Questions