Class 8 Maths · Original EZT practice
Squares and Cubes (Class 8): original worked practice
A pattern can reject a wrong answer before a long calculation begins. Keep the integer domain, the unit digit and the factor count beside each square or cube before trusting a shortcut.
8 original worked examples, 8 fresh self-check questions and 4 mixed questions, with explained answers. These are not copied textbook exercises, an official chapter index or a full answer key. No affiliation with NCERT is claimed.
Official sources for topic and edition context, checked 17 October 2026:
Choose, solve, check, then try a fresh question.
Before calculating, read the method prompt and name the condition that makes it useful. Write your own attempt. Open the steps only when you need them, and use the check to catch a wrong sign, unit or interpretation. Finish with the fresh question before reading its answer.
See the next square through an odd gap
Choose the method: Use (n + 1)^2 = n^2 + (2n + 1) when one square is already known.
EZT practice question
A square grid has 28^2 dots. How many more dots are needed for a 29 by 29 grid, and what is 29^2?
Show the worked solution
- The gap after 28^2 is the next odd number: 2 × 28 + 1 = 57.
- 28^2 = 784, so 29^2 = 784 + 57 = 841.
Answer: 57 more dots; 29^2 = 841
Check: 29 × 29 = 841, so the odd-number gap gives the same result.
Watch for this: Do not add 2n instead of the full odd gap 2n + 1.
Now try it yourself
A pattern has reached 41^2 = 1,681. Use the odd gap to find 42^2.
Check your answer
42^2 = 1,764
2 × 41 + 1 = 83, and 1,681 + 83 = 1,764.
Use a unit digit to rule out a square
Choose the method: For a nonnegative integer, a perfect square can end only in 0, 1, 4, 5, 6 or 9.
EZT practice question
Can 8,173 be a perfect square? Give the fastest reason.
Show the worked solution
- Its unit digit is 3.
- No nonnegative integer square ends in 3, so no further multiplication is needed.
Answer: No, because it ends in 3
Check: Squaring each possible unit digit 0 through 9 gives only 0, 1, 4, 5, 6 or 9.
Watch for this: An allowed unit digit is only a pass to keep checking; it is not proof that the number is a square.
Now try it yourself
Can 6,758 be a perfect square?
Check your answer
No, because it ends in 8
8 is not a possible unit digit of an integer square.
Square a number ending in 5
Choose the method: For 10a + 5, multiply a by a + 1 and attach 25, because (10a + 5)^2 = 100a(a + 1) + 25.
EZT practice question
Use the ending-in-5 pattern to calculate 75^2.
Show the worked solution
- Here a = 7, so a(a + 1) = 7 × 8 = 56.
- Attach 25 to get 5,625.
Answer: 5,625
Check: 75 × 75 = 5,625.
Watch for this: This pattern is for a number whose last two digits are 05, 15, 25 and so on, not for every number containing a 5.
Now try it yourself
Calculate 95^2 with the same pattern.
Check your answer
9,025
9 × 10 = 90, then attach 25.
Double the trailing zeros in a square
Choose the method: Each trailing zero in a nonzero whole number contributes a factor of 10, whose square contributes 100.
EZT practice question
A nonzero whole number ends in exactly 4 zeros. How many zeros will its square end in?
Show the worked solution
- The number can be written as a × 10^4, where a does not end in zero.
- Its square is a^2 × 10^8, so it ends in 8 zeros.
Answer: 8 zeros
Check: For example, 2,300 has exactly 2 trailing zeros, and 2,300^2 = 5,290,000 has exactly 4.
Watch for this: Count only the trailing zeros of the starting number; nonzero digits can affect the digits before them.
Now try it yourself
A nonzero whole number ends in exactly 3 zeros. How many zeros will its square end in?
Check your answer
6 zeros
Squaring doubles the exact trailing-zero count.
Prove a square from prime factors
Choose the method: A positive integer is a perfect square when every prime factor has an even exponent.
EZT practice question
Is 3,364 a perfect square? If yes, find its positive square root.
Show the worked solution
- 3,364 = 4 × 841 = 2^2 × 29^2.
- All exponents are even, so 3,364 = (2 × 29)^2 = 58^2.
Answer: Yes; the positive square root is 58
Check: 58^2 = 3,364.
Watch for this: Do not stop because 841 looks unfamiliar; factor it as 29^2.
Now try it yourself
Is 2,704 a perfect square? If yes, find its positive square root.
Check your answer
Yes; 52
2,704 = 2^4 × 13^2 = (2^2 × 13)^2 = 52^2.
Bracket a square root before estimating
Choose the method: Place the number between two known consecutive squares, then compare distances to decide which side it is nearer.
EZT practice question
Between which two consecutive whole numbers does √1,700 lie, and which is it nearer?
Show the worked solution
- 41^2 = 1,681 and 42^2 = 1,764.
- 1,700 is between them; it is 19 above 1,681 and 64 below 1,764, so it is nearer 41.
Answer: Between 41 and 42, nearer 41
Check: 41.2^2 = 1,697.44, close to 1,700, so the estimate is reasonable.
Watch for this: The midpoint of the roots is not automatically the square root of the midpoint of the squares.
Now try it yourself
Between which two consecutive whole numbers does √2,300 lie, and which is it nearer?
Check your answer
Between 47 and 48, nearer 48
47^2 = 2,209 and 48^2 = 2,304; 2,300 is only 4 below 2,304.
Predict a cube's unit digit and parity
Choose the method: Only the unit digit of the base affects the unit digit of its cube, and even or odd is preserved by cubing.
EZT practice question
What is the unit digit of 68^3, and is the cube even or odd?
Show the worked solution
- The base ends in 8, and 8^3 = 512, so the cube ends in 2.
- 68 is even, so its cube is even.
Answer: Unit digit 2; even
Check: 68^3 = 314,432, which ends in 2 and is even.
Watch for this: Unlike squares, cubes can end in any digit, so use the digit's cube rather than rejecting the number.
Now try it yourself
What is the unit digit of 53^3, and is it even or odd?
Check your answer
Unit digit 7; odd
3^3 = 27, and 53 is odd; 53^3 = 148,877.
Complete the prime-factor groups for a cube
Choose the method: A positive integer is a perfect cube when every prime exponent is a multiple of 3.
EZT practice question
Find the smallest positive integer by which 1,176 must be multiplied to become a perfect cube.
Show the worked solution
- 1,176 = 2^3 × 3 × 7^2.
- The 2s are complete; add 3^2 and one more 7 to complete their groups.
- The multiplier is 3^2 × 7 = 63, and 1,176 × 63 = 74,088 = 42^3.
Answer: 63
Check: 42^3 = 74,088.
Watch for this: Do not count prime factors; compare each exponent with the next multiple of 3.
Now try it yourself
Find the smallest positive integer by which 1,323 must be multiplied to become a perfect cube.
Check your answer
7
1,323 = 3^3 × 7^2, so one more 7 gives 9,261 = 21^3.
Try a mixed set
Choose a method or explanation before opening each answer.
1. Is 0 a perfect square and a perfect cube?
Show answer and reason
Yes, in the whole-number or integer domain.
0 = 0^2 and 0 = 0^3. The domain matters because some textbook lists begin with natural numbers.
2. Can a negative integer be a perfect cube?
Show answer and reason
Yes.
For example, (-5)^3 = -125. A negative integer cannot be the square of an integer.
3. Why is 1,729 called a taxicab number?
Show answer and reason
It is the smallest positive integer that is a sum of two positive cubes in two different ways.
1,729 = 1^3 + 12^3 = 9^3 + 10^3. The positive-integer condition is part of the claim.
4. A number ends in 6. Must it be a perfect square?
Show answer and reason
No.
6 is allowed for a square, but 26 ends in 6 and is not a perfect square. Unit-digit screening is necessary for rejection, not sufficient for proof.
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Questions about this practice page
- Why do perfect squares never end in 2, 3, 7 or 8?
- For nonnegative integers, the square's unit digit depends only on the original unit digit. Squaring 0 through 9 produces only 0, 1, 4, 5, 6 or 9.
- Does a correct unit digit prove a number is a perfect square?
- No. It only means the number has not been ruled out. Use factorisation, a known square or another check to prove it.
- What is the fastest way to square a number ending in 5?
- For 10a + 5, multiply a by a + 1 and attach 25. For example, 85^2 uses 8 × 9 = 72, so the result is 7,225.
- What is special about 1,729?
- It is the smallest positive integer expressible as a sum of two positive cubes in two ways: 1^3 + 12^3 and 9^3 + 10^3.
- Are these the NCERT Ganita Prakash exercise questions?
- No. These are original EZT practice questions on this chapter's ideas. Use them alongside your textbook, not as a replacement for it.