How many practice questions are available for Squares and Square Roots?

SparkEd Math offers 60+ practice questions for Squares and Square Roots across 3 difficulty levels (Easy, Medium, Hard) plus a timed test mode. Each question comes with visual step-by-step solutions and AI coaching hints.

Is Squares and Square Roots content free on SparkEd Math?

Yes! All Squares and Square Roots content on SparkEd Math is 100% free, including practice questions, visual solutions, lessons, and AI coaching. No credit card required.

How does SparkEd Math teach Squares and Square Roots differently from textbooks?

SparkEd Math uses visual step-by-step solutions for every Squares and Square Roots problem, making complex concepts easy to understand. The platform includes 3 difficulty levels for progressive learning, a Super Power Help AI coach for instant doubt-clearing, and interactive practice rather than passive reading.

Is SparkEd's Squares and Square Roots content aligned with the Olympiad syllabus?

Yes, all Squares and Square Roots questions and lessons on SparkEd Math are fully aligned with the Mathematical Olympiad Preparation (Olympiad) syllabus for Class 8. The content follows the official curriculum structure and exam pattern.

Squares and Square Roots — Class 8 Olympiad (2025-26)

About Squares and Square Roots — Class 8 Olympiad

Find squares and square roots using various methods; solve Olympiad problems involving perfect squares and patterns. This topic is part of the Olympiad Class 8 mathematics syllabus (chapter: Module 3). On this page you can practice 60 questions across three difficulty levels — 20 easy, 20 medium, and 20 hard — each with a visual step-by-step solution, plus a timed 35-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

Squares and Square Roots — solved examples for Class 8 Olympiad

Example 1easy

Which of the following numbers CANNOT be a perfect square?

A)A) 12345676
B)B) 98765432
C)C) 56789001
D)D) 23456789

Step-by-step solution

The unit digit of a perfect square can only be 0, 1, 4, 5, 6, or 9.
Numbers ending in 2, 3, 7, or 8 can never be perfect squares.
Option B, 98765432, ends in 2. Therefore, it cannot be a perfect square.

Answer: B) 98765432

Example 2medium

A natural number N ends with the digit 5. If N is squared, its result (N²) has last two digits that are not both zero. What is the tens digit of N²?

Step-by-step solution

A natural number N ending with 5 can be written in the form N = 10k + 5 for some integer k ≥ 0.
Squaring N, we get N² = (10k + 5)² = (10k)² + 2(10k)(5) + 5² = 100k² + 100k + 25.
Factoring out 100 from the first two terms: N² = 100(k² + k) + 25.
This expression shows that N² will always end in 25. Therefore, the tens digit of N² is always 2.

Answer: 2

Example 3hard

If N is a perfect square and its last digit is 6, which of the following statements about its tens digit (T) must be true?

A)A. T is an even digit.
B)B. T is an odd digit.
C)C. T is either 0 or 2.
D)D. T is either 4 or 6.

Step-by-step solution

A perfect square ending in 6 must be the square of a number whose unit digit is either 4 or 6.
Case 1: Numbers ending in 4. Let the number be (10k + 4). Its square is (10k + 4)² = 100k² + 80k + 16. The tens digit is determined by (80k + 16), which means the tens digit is the unit digit of (8k + 1), which is always odd (e.g., 16, 196, 576, ...).
Case 2: Numbers ending in 6. Let the number be (10k + 6). Its square is (10k + 6)² = 100k² + 120k + 36. The tens digit is determined by (120k + 36), which means the tens digit is the unit digit of (12k + 3), which is always odd (e.g., 36, 256, 676, ...).
In both cases, the tens digit of a perfect square ending in 6 must be an odd digit.

Answer: B. T is an odd digit.

Practice questions on Squares and Square Roots

Q1.easy
If S = 1 + 3 + 5 + ... + (2n-1), and S = 1024, what is the value of n?
1. A)A) 32
2. B)B) 16
3. C)C) 64
4. D)D) 1024
Show answer
Answer: A) 32
Hint: Remember the special property related to the sum of the first 'n' odd natural numbers.
Q2.easy
What is the smallest positive integer by which 1575 must be multiplied so that the product is a perfect square?
1. A)A) 3
2. B)B) 5
3. C)C) 7
4. D)D) 15
Show answer
Answer: C) 7
Hint: Perform the prime factorization of 1575 and identify any prime factors that do not appear in pairs.
Q3.easy
Which of the following is the best approximation for the value of √(300 × 301 × 302 × 303 + 1)?
1. A)A) 90900
2. B)B) 90901
3. C)C) 90902
4. D)D) 90903
Show answer
Answer: B) 90901
Hint: Consider the algebraic pattern for the product of four consecutive integers plus one. Let the smallest integer be 'n'.
Q4.medium
What is the smallest natural number by which the product of the first six even natural numbers must be multiplied to make it a perfect square?
1. A)5
2. B)3
3. C)10
4. D)15
Show answer
Answer: 5
Hint: First, write down the product of the first six even numbers. Then, find the prime factorization of this product.
Q5.medium
Consider the sequence S_n = 1 + 3 + 5 + ... + (2n-1). If S_k + S_(k+1) = 225, what is the value of k?
1. A)6
2. B)7
3. C)8
4. D)9
Show answer
Answer: 7
Hint: Recall the formula for the sum of the first 'n' odd natural numbers.
Q6.medium
If the square of a natural number has 'm' digits, and its square root has 'n' digits, which of the following statements is always true?
1. A)m = 2n-1
2. B)m = 2n
3. C)2n-1 ≤ m ≤ 2n
4. D)m ≥ 2n
Show answer
Answer: 2n-1 ≤ m ≤ 2n
Hint: Consider the range of values for a number with 'n' digits and its square. Test with small examples like N=3, N=10, N=31, N=32.
Q7.hard
How many 4-digit perfect squares are there such that the sum of their digits is 19 and their units digit is 9?
1. A)A. 1
2. B)B. 2
3. C)C. 3
4. D)D. 4
Show answer
Answer: C. 3
Hint: A 4-digit perfect square has a square root between 32 and 99. If its units digit is 9, its square root must end in 3 or 7. Systematically check numbers matching these criteria.
Q8.hard
Find the smallest natural number 'k' such that 1470 × k is a perfect square, and 1470 / k is also a perfect square.
1. A)A. 30
2. B)B. 70
3. C)C. 105
4. D)D. 210
Show answer
Answer: A. 30
Hint: First, find the prime factorization of 1470. For a number to be a perfect square, all exponents in its prime factorization must be even. Consider how 'k' must affect the odd exponents.
Q9.hard
What is the value of √[ (1 + 3 + 5 + ... + 39) ]?
1. A)A. 10
2. B)B. 20
3. C)C. 40
4. D)D. 400
Show answer
Answer: B. 20
Hint: Recall the fundamental property that the sum of the first 'n' consecutive odd natural numbers is n². Determine how many odd numbers are present in the sum 1 + 3 + 5 + ... + 39.

These are 9 of the 60 questions available for Squares and Square Roots. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.