Chapter 5: Taking Square Root F.2 Mathematics Enrichment Taking Square Root Taking Square Root of a number can be done easily by using a calculator in normal lessons. How about when there is no calculator? This is common in competitions. Do we have any way to find the square root without using a calculator? 1. Simple Square Numbers 12 = 62 = 112 = 162 = 2. 22 = 72 = 122 = 172 = 32 = 82 = 132 = 182 = 42 = 92 = 142 = 192 = 52 = 102 = 152 = 202 = Square Root of a Large Square Number Find 676 (Given that 676 is an integer). Step 1: Find the range of the square root. As 252 = 625 and 302 = 900, 25  676  30 Step 2: Find the unit digit of the square root. Unit digit of the square number Unit digit of the square root 0 1 4 9 6 5 6 9 4 1 0 1 2 3 4 5 6 7 8 9 From the table, 676  26 e.g.1: Given that 1764 and 3249 are both square numbers, find their square roots. 1600 = __________ 2500 = __________ _52 = ___25 2500 = __________ 3600 = __________ _52 = ___25 _____ < 1764 < ______ _____ < 3249 < ______ By considering the unit digits, 1764  __________ 3249  __________ Page 1 Chapter 5: Taking Square Root F.2 Mathematics Enrichment 3. Square Root of Any Degree of Accuracy Find Number with a given 650.5 . (correct to the nearest integer) Step 1: Find the range of the square root. As 252 = 625 and 262 = 676, 25  650.5  26 Hence, 650.5 is either 25 or 26. Step 2: Find the square of the average of two numbers got in Step 1. 25.52 = 650.25, as 650.25 < 650.5, 25.5  650.5  26 As a result, 650.5  26 (correct to the nearest integer) e.g.2: Find the square root of 1234. (correct to the nearest integer) 900 = __________ 1600 = __________ 1225= __________ 362 = __________ Therefore, __________  1234  __________ __________2 = __________ __________  1234  __________ As a result, 1234  __________ e.g.3: Find the square root of 56.78. (correct to 1 decimal place) 49 = __________ 64 = __________ 56.25 = __________ 7.62 = __________ Therefore, __________  56.78  __________ __________2 = __________ __________  56.78  __________ As a result, 56.78  __________ Page 2 Chapter 5: Taking Square Root F.2 Mathematics Enrichment 4. “Long Division” to Find Square Root Find 650.5 . (correct to the nearest integer) 2 6 - 4 2 2 50. 50 Step 1: Separate the number into groups of 2 digits (starting from the decimal point): 6, 50, 50 (add a zero following the last ‘5’). 50 Step 2: Look for an integer such that its square is just smaller than the first group (6), and the integer is 2. Now we have a remainder 250. 2 4__ 2 45 2 6 - 4 2 __. 50. 2 6 - 4 2 - 2 5. 50. 50 50 25 25 50 50 Step 3: Then double what we have in the result (2 x 2 = 4). 50 Step 4: Fill in the blanks in step 3 by the same digit such that the product of (4x) and x is just smaller than the remainder 250. The digit should be 5. Step 3 and 4 can then be repeated to obtain the required degree of accuracy. (00 may be added if the next group of two digits is needed) 2 45 50__ 2 6 - 4 2 - 2 5. 50. 50 25 25 __ 50 2  50 45 2 6 - 4 2 - 2 505 - 5. 50. 50 25 25 25 5 50 50 25 25 650.5  25.5...  650.5  26 (correct to the nearest integer) Use the “Long Division” method to find the following square roots. e.g.4: Find the square root of 1234. (correct to the nearest integer) e.g.5: Find the square root of 56.78. (correct to 1 decimal place) Page 3 Chapter 5: Taking Square Root F.2 Mathematics Enrichment 5. How Does The “Long Division” Work? Remember the identity (a + b) 2 = a2 + 2ab + b2? To explain the method, we change the identity a little bit: (10a + b)2 = 100a2 + 20ab + b2 Take 676 as an example. a 6 a 2 46 2 6 - 4 2 - 2 We want to find the largest integers a, b such that (10a + b)2 76 is just less than or equal to 676. We test for a first, obviously, the largest possible value is 2. 6 76 Then what’s left = 676 – 400 = 276 Now, we have to try the greatest b such that 20(2)b + b2 = (40 +b) b is just less than or equal to 276 76 76 As 46(6) = 276, 676  26 Use methods in section 3 or 4 to find the following square roots. (correct to 2 sig. fig.) 2 38 50 35.9 Page 4 Chapter 5: Taking Square Root F.2 Mathematics Enrichment Homework Solve the following questions. 1. Given that 7921 and 15129 are square numbers, find their square roots. 2. Use methods in section 3 or 4 to find the following square roots. (correct to 2 sig. fig.) (a) 12345 (b) 187.69 (c) 35.835 3. The figure shows an acute-angled triangle with area 120. Find the integer closest to x. [6th Pui Ching Invitational Mathematics Competition Heat Event (Secondary 2), no.13] Hint: Cut the triangle into two right-angled triangles by adding a suitable height Page 5