Firstly, we use "The First by the First and the Last by the Last" technique to solve the square root. (1). √6889 There are two groups of figures, '68' and '89'. So we expect 2-digit answer. Now see since 68 is greater than 64(82) and less than 81(92), the first figure must be 8. So, 6889 is between 6400 and 8100, that means, between 802 and 902. Now look at the last figure of 6889, which is 9. Squaring of numbers 3 and 7 ends with 9. So, either the answer is 83 or 87. There are two easy ways of deciding. One is to use the digit sums. If 872 = 6889 Then converting to digit sums (L.H.S. is 8+7 = 15 -> 1+5 -> 6 and R.H.S. is 6+8+8+9 -> 31 -> 3+1 -> 4) We get 62 -> 4, which is not correct. But 832 = 6889 becomes 22 -> 4, so the answer must be 83. The other method is to recall that since 852 = 7225 and 6889 is below this. 6889 must be below 85. So it must be 83. Note: To find the square root of a perfect 4-digit square number we find the first figure by looking at the first figures and we find two possible last figures by looking at the last figure. We then decide which is correct either by considering the digit sums or by considering the square of their mean. (2). √5776 The first 2-digit(i.e. 57) at the beginning is between 49 and 64, so the first figure must be 7. The last digit (i.e. 6) at the end tells us the square root ends in 4 or 6. So the answer is 74 or 76. 742 = 5776 becomes 22 -> 7 which is not true in terms of digit sums, so 74 is not the answer. 762 = 5776 becomes 42 > 16 -> 7, which is true, so 76 is the answer. Alternatively to choose between 74 and 76 we note that 752 = 5625 and 5776 is greater than this so the square root must be greater than 75. So it must be 76.

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