As a mathematical concept, the square root is one of the most critical skills that students need to understand. It is essential in many fields and branches of study, including science, engineering, and finance. In this post, we will focus on calculating the square root of 74.

The calculation of square roots can be performed using different methods. One of the most straightforward methods is the long division method, which involves repetitive guessing and checking until the result is obtained with reasonable accuracy. Here are the steps to find the square root of 74:

Step 1: Divide 74 into two-digit pairs from right to left, starting with the units digit. In this example, 74 has only one digit, so add a zero to make it an even number. The two-digit pairs then become (7, 40).

Step 2: Find the largest square number less than or equal to the first pair (7). The largest square number that fits this category is 4^2=16. Write down the 4 as the first digit of the square root, then subtract 16 from 7 to get 7-16=-9.

Step 3: Bring down the next pair (40) to the right of the remainder (−9) to form 940.

Step 4: Twice the square root number from step 2 (in this case, 4) is 8. Write this number on top of the new dividend: 940.

Step 5: Find the largest number that, when placed as the next digit in the divisor, makes the product less than or equal to 94008. That number is 5 if we multiply 845 by 5, we get 4225.

Step 6: Subtract 4225 from 94008. We get 89783.

Step 7: Bring down the next pair (00) to the right of the remainder (89783) to form 897830.

Step 8: Double the square root number (in this case, 45) to get 90. Write this number on top of the new dividend: 897830.

Step 9: Find the largest number that, when placed as the next digit in the divisor, makes the product less than or equal to 897830*10+0. That number is 9 if we multiply 9015 by 9, we get 81135.

Step 10: Subtract 81135 from 8978300. We get 8897165.

Step 11: Bring down the next pair (00) to the right of the remainder (8897165) to form 889716500.

Step 12: Double the square root number (in this case, 450) to get 900. Write this number on top of the new dividend: 889716500.

Step 13: Find the largest number that, when placed as the next digit in the divisor, makes the product less than or equal to 889716500*10+0. That number is 9 if we multiply 90050 by 9, we get 810450.

Step 14: Subtract 810450 from 8897165000. We get 8896354550.

Step 15: Bring down the next pair (00) to the right of the remainder (8896354550) to form 889635455000.

Step 16: Double the square root number (in this case, 4500) to get 9000. Write this number on top of the new dividend: 889635455000.

Step 17: Find the largest number that, when placed as the next digit in the divisor, makes the product less than or equal to 889635455000*10+0. That number is 3 if we multiply 900032 by 3, we get 2700096.

Step 18: Subtract 2700096 from 8896354550000. We get 8896351849904.

The result shows that the square root of 74 is approximately 8.602. Although this method may seem tedious, it is an essential skill to have, particularly when dealing with large numbers or solving problems in calculus and other advanced mathematics.