Practice Questions

Calculate the value of $(1.1)^3$.

List the cubes of the integers from 1 to 5.

Calculate the cube root of 4096 using the prime factorization method.

Find the one's digit of the cube of the number 789.

Evaluate if a perfect cube can end with exactly two zeros. Justify your reasoning.

State the symbol used to denote the cube root of a number.

Identify the one's digit of the cube of the number 129.

What is the cube of 7?

Define a perfect cube or a cube number.

Justify why 27000 is a perfect cube but 2700 is not, by analyzing the number of trailing zeros.

Evaluate the statement: 'The cube of any integer ending in 7 will end in 3.' Justify your answer.

Find the smallest natural number by which 3087 must be multiplied so that the product is a perfect cube.

Find the smallest natural number by which 10985 must be divided to obtain a perfect cube.

Describe the pattern of the one's digit for the cubes of numbers ending in 2 and 8.

A student claims that to find the smallest number to multiply 1080 by to make it a perfect cube, you only need to multiply by the factors that are not in a triplet. Critique this method and then justify the correct smallest number.

A large cube has a volume represented by the expression $216x^6y^3$ cubic units. Formulate an expression for the length of one side of this cube and justify your process.

Design a three-digit number which is a perfect cube and where the sum of its digits is also a perfect cube. Justify your selection.

Evaluate the statement: 'The cube of a two-digit number can have at most six digits.' Justify your conclusion by creating arguments based on the smallest and largest two-digit numbers.

Evaluate the pattern $n^3 - (n-1)^3 = 1 + 3n(n-1)$. Justify if this pattern holds true for all integers $n$. Use it to find the value of $15^3 - 14^3$.

Three numbers are in the ratio $1:2:3$. The sum of their cubes is $972$. Solve for the numbers.

Find the cubes of the following numbers: (a) 11 (b) 1.2

Solve for the integer $y$ if $y^3 = -512$.

Calculate the value of $\sqrt[3]{64 \times 729}$.

Explain the relationship between the operations of finding a cube and finding a cube root.

List all the perfect cubes between 1 and 300. Also, identify which of them are cubes of even numbers and which are cubes of odd numbers.

Explain the process of finding the cube root of a perfect cube using the prime factorization method. Use the number 1728 as an example to illustrate the steps.

Identify which of the following numbers are perfect cubes and explain your reasoning for each using prime factorization: (i) 216 (ii) 500 (iii) 729 (iv) 1000 (v) 343

Examine if 2700 is a perfect cube. Justify your answer.

The volume of a cubical container is $2.197 \text{ m}^3$. Calculate the length of its edge.

Calculate the value of $\sqrt[3]{-1331} \times \sqrt[3]{343}$.

Critique the statement: 'If a number is a perfect square, it cannot be a perfect cube.' Justify your critique with a counterexample.

Formulate a general expression for the cube of any even number and justify why the result must always be an even number.

Recall the first Hardy-Ramanujan Number and state why it is special.

Summarize the rule for a number to be a perfect cube based on its prime factorization.

Explain why 100 is not a perfect cube by using the prime factorization method.

Formulate a general rule for the one's digit of the cube of any integer. Create a table to present your findings and justify the pattern for numbers ending in 2, 4, and 8.

Justify that the difference between the cubes of two consecutive positive integers, $n$ and $(n+1)$, is never divisible by 3.

A metal cuboid has dimensions $18 \text{ cm} \times 24 \text{ cm} \times 27 \text{ cm}$. This cuboid is melted and recast into smaller cubes of edge length $3 \text{ cm}$. Calculate the number of smaller cubes that can be formed.

Describe the property related to the sum of consecutive odd numbers that results in a perfect cube. Provide the sums for $1^3$, $2^3$, and $3^3$ to illustrate this pattern.

A school decides to build a cubical water tank whose volume is $5832 \text{ m}^3$. They need to paint the outer surface of the tank, excluding the base. If the cost of painting is $₹ 150$ per square meter, calculate the total cost of painting the tank.

Design an algorithm (a set of steps) to determine the smallest natural number by which a given number $N$ must be divided to make the quotient a perfect cube. Justify each step. Apply your algorithm to the number 1188.

Analyze the pattern $n^3 - (n-1)^3 = 1 + n \times (n-1) \times 3$. Apply this pattern to calculate the value of $15^3 - 14^3$.

Demonstrate that 4104 is a Hardy-Ramanujan number by showing it can be expressed as the sum of two cubes in two different ways, using the pairs of numbers $(2, 16)$ and $(9, 15)$.

Critique the following argument: 'The number 1729 is the smallest number expressible as the sum of two cubes in two different ways. Therefore, there are no numbers smaller than 1729 that can be written as the sum of two cubes.' Justify your critique with examples.

Propose a method to estimate the cube root of a 5-digit perfect cube, for example 15625, by analyzing its digits. Justify the steps.