Practice Questions

List the possible digits in the units place of a perfect square number.

What is the value of $5^3$?

Calculate the square of $1.5$.

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

Define a perfect square.

Identify which of the following numbers cannot be a perfect square, based on its units digit: 144, 169, 197, 225.

State the property of a perfect square related to the number of zeros at the end.

Identify the square of each of the following numbers: (i) 11 (ii) 15 (iii) 20.

Explain why a number ending in the digit 2, 3, 7, or 8 cannot be a perfect square.

List all the perfect squares that are between 50 and 150.

A square-shaped garden has an area of $729 \text{ m}^2$. Calculate the length of the fence required to enclose the garden completely.

Examine the number 7744. Without calculating the square root, determine the units digit of its square root.

Examine the pattern: $1^2 = 1$, $11^2 = 121$, $111^2 = 12321$. Apply this pattern to find the square of $11111$.

Justify why a perfect square cannot end in the digit 7, using the concept of the unit digits of numbers from 0 to 9.

Propose a number greater than 1 that is both a perfect square and a perfect cube. Justify your choice using prime factorization.

Summarize the prime factorization method used to identify if a number is a perfect cube.

Justify algebraically why the sum of the first 'n' odd natural numbers is $n^2$.

Critique the statement: "If the sum of the digits of a number is a perfect square, the number itself must be a perfect square." Provide a counterexample to support your critique.

Design a 3-step method to find the cube root of a 5-digit perfect cube (e.g., 32768) without using prime factorization. Your method should rely on analyzing the unit digit and the remaining digits.

Without performing the full calculation, state the units digit of the square of each of the following numbers: (i) 27 (ii) 109 (iii) 34.

Explain the difference between a square number and a cube number. Provide two examples of perfect squares, two examples of perfect cubes, and one example of a number that is both a perfect square and a perfect cube.

Apply the property of successive odd numbers to demonstrate that 49 is a perfect square.

A general wishes to arrange his 8289 soldiers in the form of a square. After arranging them, he found that 8 soldiers were left out. Calculate how many soldiers were there in each row of the square formation.

Evaluate the claim: "The cube of a two-digit number can never have more than six digits." Justify your conclusion by considering the smallest and largest two-digit numbers.

Justify why the difference between the squares of two consecutive integers, $(n+1)^2 - n^2$, is always an odd number. Use this property to find $51^2$ if you know $50^2 = 2500$.

Formulate a general rule for the number of non-perfect square integers between two consecutive perfect squares, $n^2$ and $(n+1)^2$. Justify your rule.

Describe the method of repeated subtraction to determine if a number is a perfect square. Use this method to find the square root of 49.

Explain the term 'cube root'. Then, find the cube root of 216.

Describe the relationship between perfect squares and the sum of consecutive odd numbers starting from 1. Show with examples for the first four perfect squares.

Evaluate which method is more efficient for determining if 729 is a perfect square: repeated subtraction of odd numbers or prime factorization. Justify your reasoning.

Calculate the value of $\sqrt{176 + \sqrt{2401}}$.

Compare the number of zeros at the end of $800$ and its square, $800^2$.

Solve for the smallest whole number by which 392 must be multiplied to get a perfect square. Also, calculate the square root of the new number.

The volume of a cubical box is $5832 \text{ cm}^3$. Calculate the length of one edge of the box.

Analyze the number 21952. Use prime factorization to determine if it is a perfect cube. If yes, calculate its cube root.

A school collected ₹2401 from its students for a charity fund. Each student donated as many rupees as the number of students in the school. Calculate the number of students in the school and the amount donated by each student.

List the cubes of the first 10 natural numbers. From your list, identify and describe the pattern of the units digits of the cubes.

The locker puzzle in the source text results in lockers with square numbers remaining open. Formulate a new rule for the puzzle such that only the lockers with numbers that are perfect cubes (up to 1000) will remain open. Describe the actions of Person 2 and Person 3 under your new rule.

Three metallic cubes with edges 3 cm, 4 cm, and 5 cm respectively are melted and recast into a single, larger cube. Calculate the edge of the new cube formed.

Formulate a hypothesis about the relationship between the number of digits in a number ($d$) and the number of digits in its cube. Test your hypothesis for 1-digit, 2-digit, and 3-digit numbers, and justify your conclusion.

Solve for the smallest number that should be subtracted from 1989 to make it a perfect square. Also, calculate the square root of the resulting perfect square.

The sum of two cubes can be factored as $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$. Use this formula to verify that 1729 can be expressed as the sum of two cubes in two different ways ($1^3 + 12^3$ and $9^3 + 10^3$). Then, critique why this factorization alone is not sufficient to find such "taxicab" numbers easily.

A general has to arrange his soldiers, numbering 8200, in the form of a solid square. After arranging them, he found that some soldiers were left out. Design a process to calculate how many soldiers were left out. Then, propose the minimum number of additional soldiers required to form the next larger solid square.

Prove that for any natural number $n > 1$, the triplet $(2n, n^2-1, n^2+1)$ forms a Pythagorean triplet. Then, create three different Pythagorean triplets using this formula.

Create a problem where one must find the smallest number to multiply 1080 by to make it a perfect square, and then find the smallest number to divide the result by to make it a perfect cube. Solve the problem you have created.