Revision and important questions: A Square and a Cube

5.1 Formula box

A perfect square never ends in 2, 3, 7 or 8. Its trailing zeros are even in number. A perfect cube can end in any digit, and its trailing zeros are a multiple of three. A number is a perfect square when its prime factors divide into two matching groups; it is a perfect cube when they divide into three.

5.2 Choosing the right method

For a small number, listing nearby squares is fastest. For a large perfect square, prime factorisation is the most reliable method because it also proves whether the number is a square. Repeated subtraction of odd numbers is useful for showing why a square works, but it becomes slow beyond a few hundred. Estimation with the last digit is the best first move for a large number whose square status is not known.

The same logic applies to cubes. Prime factorisation into triplets gives a proof; the last digit gives a fast guess. Estimate the tens digit by placing the number between two known cubes, then verify by cubing.

5.3 Quick self-check

A good answer to a square question usually has three parts: the method, the arithmetic, and a check. For example, to show that 1764 is a square, state the method (factorisation or interval), carry out the steps, and finish by squaring the answer. If the squared answer returns the original number, the work is consistent.

Keep a small checklist while practising: did I use the correct formula, did I keep the digits in place, and did I verify the final answer? These three questions catch most slips in this chapter. Write each step on its own line so that a small arithmetic error is easy to spot.

5.4 Practice routine

Spend ten minutes a day on mixed questions: two squares patterns, one square root by factorisation, one cube root by estimation, and one word problem. Change the numbers rather than the method. When a method feels automatic, write the reasoning in one sentence; that sentence is exactly what an exam answer needs.

5.5 Common mistakes

• Thinking any number that ends with zero, one, four, five, six or nine must be a square. The units digit can only rule a number out; it cannot confirm a square.
• Forgetting the negative square root. Every positive perfect square has two integer roots, though this chapter uses the positive one.
• Pairing prime factors wrongly. In 2^6 x 3 x 7^2, the unpaired 3 has to be fixed; do not pair a 3 with a 7.
• Counting the endpoints when asked how many numbers lie between two squares. Between 16^2 and 17^2 there are 32 numbers, not 33 or 34.
• Treating cube roots like square roots. A cube root is unique for a perfect cube; there is no separate negative root in the same way.
• Guessing cube roots only from the last digit. The last digit narrows the choice, but the size of the number decides between candidates ending in the same digit.
• Assuming the difference between consecutive cubes is odd and constant. It is odd but grows each step.
• Writing 2^3 as 6. The exponent tells how many copies of the base are multiplied, so 2^3 = 8.

5.6 Important questions with answers

Question 1. Find the square root of 1764.

Answer. Since 40^2 = 1600 and 42^2 = 1764, the root is 42. One can also factorise 1764 = 2^2 x 3^2 x 7^2 = (42)^2.

Question 2. Is 2116 a perfect square?

Answer. Yes. Estimate between 45^2 = 2025 and 47^2 = 2209; test 46^2 = 2116. So the square root is 46.

Question 3. Find the smallest number by which 720 must be multiplied to make a perfect square.

Answer. 720 = 2^4 x 3^2 x 5. The 5 is unpaired, so multiply by 5: 3600 = 60^2. The multiplier is 5.

Question 4. Find the smallest number by which 1800 must be divided to make a perfect square.

Answer. 1800 = 2^3 x 3^2 x 5^2. The 2 needs its exponent reduced from 3 to 2, so divide by 2: 900 = 30^2. The divisor is 2.

Question 5. Find the cube root of 13824.

Answer. 13824 = 8 x 1728 = 2^3 x 12^3 = 24^3, so the cube root is 24.

Question 6. What is the smallest number by which 243 must be multiplied to make a perfect cube?

Answer. 243 = 3^5, so one more 3 makes 3^6 = 9^3 = 729. The multiplier is 3.

Question 7. How many numbers lie between 30^2 and 31^2?

Answer. Between n^2 and (n+1)^2 there are 2n numbers, so between 900 and 961 there are 60 numbers.

Question 8. Find the difference between 21^2 and 22^2.

Answer. The difference between consecutive squares is their sum: 21 + 22 = 43. Check: 484 - 441 = 43.

Question 9. Evaluate 37^3 - 36^3.

Answer. Use a^2 + ab + b^2 with 37 and 36: 1369 + 1332 + 1296 = 3997. So 37^3 - 36^3 = 3997.

Question 10. Without computing, decide whether 2025 is a perfect square.

Answer. It ends in 25, which is allowed, and 45^2 = 2025. So it is a perfect square, and its square root is 45.