Squares and Square Roots

A natural number m is a perfect square if m = n^2 for some natural number n, as with 1, 4, 9, 16 and 25. In this chapter you will discover the properties of square numbers: they end only in 0, 1, 4, 5, 6 or 9, they have an even number of zeros at the end, the square of an odd number is odd, and there are 2n non-square numbers between n^2 and (n + 1)^2. You will see interesting patterns, such as the sum of the first n odd numbers being n^2, and learn that 2m, m^2 - 1 and m^2 + 1 always form a Pythagorean triplet. The second part deals with square roots, the inverse operation of squaring. You will find square roots by repeated subtraction of odd numbers, by prime factorisation, pairing equal prime factors, and by the long division method, which also works for decimals such as 17.64. You will also estimate square roots of numbers that are not perfect squares and use square roots to solve problems on areas and arrangements.

Detailed Worksheet: Squares and Square Roots Section A - Definitions (10 marks) 1. What is a perfect square? Write all the perfect squares between 50 and 150. (2 marks) 2. Which digits can occur at the units place of a perfect square? Using this, explain why 2343 cannot be a perfect square. (2 marks) 3. How many non-square numbers lie between n^2 and (n + 1)^2? Use this to find the number of non-square numbers between 12^2 and 13^2. (2 marks) 4. What is a Pythagorean triplet? Write the general form of a Pythagorean triplet for any natural number m greater than 1. (2 marks) 5. Define the square root of a number. Why does 64 have two square roots, and which one is meant by its positive square root? (2 marks) Section B - Calculations and Applications (15 marks) 6. Find the square roots of the following by prime factorisation: (i) 7744 (ii) 9604 (iii) 5929. (3 marks) 7. (i) Find the smallest number by which 252 must be multiplied to get a perfect square, and find the square root of the product. (ii) Find the smallest number by which 2028 must be divided to get a perfect square, and find the square root of the quotient. (3 marks) 8. Find the square roots by the long division method: (i) 529 (ii) 3249 (iii) 7921. (3 marks) 9. Find the square roots of the decimals: (i) 2.56 (ii) 17.64 (iii) 31.36. (3 marks) 10. Write a Pythagorean triplet whose smallest member is 6, and a triplet in which one member is 14. Verify each. (3 marks) Section C - Diagrams (10 marks) 11. Draw dot patterns showing that 1 = 1^2, 1 + 3 = 2^2, 1 + 3 + 5 = 3^2, 1 + 3 + 5 + 7 = 4^2 and 1 + 3 + 5 + 7 + 9 = 5^2, shading each new L-shaped layer of dots. Explain the pattern. (4 marks) 12. Draw a right-angled triangle with sides 3 cm, 4 cm and 5 cm, and draw a square on each side. Write the area of each square and show how they illustrate a Pythagorean triplet. (3 marks) 13. Show the complete working of the long division method for finding the square root of 4489, marking the bars over the pairs of digits. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. (i) A gardener has 2025 plants. He wants to plant them so that the number of rows equals the number of plants in each row. Find the number of rows. (ii) Find the smallest square number that is divisible by each of 4, 9 and 10, explaining each step. (5 marks) 15. Without finding the squares or square roots, answer and explain: (i) What is the units digit of 1057^2? (ii) Why is 23453 not a perfect square? (iii) Why is 1000 not a perfect square? (iv) How many digits will the square roots of 4489 and 390625 have? (5 marks) 16. (i) Find the least number that must be subtracted from 1989 to get a perfect square, and find its square root. (ii) Find the least number that must be added to 1825 to get a perfect square, and find its square root. (iii) Between which two consecutive whole numbers does the square root of 250 lie, and to which is it closer? (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Show the prime factorisation or the long division working in full, and verify square roots by squaring wherever possible.