What is the root of 345 without the use of tables?
345 = 3*5*23, so no perfect squares
So, what methods of solution have you been given? The value will have to be an approximation, so use some numerical method or graphical method to get the value.
interpolation, Newton's method, bisection, or something else.
"without the use of tables" ??
- then how about just using a calculator ? ( I am being facetious)
Here is the method I actually taught in medieval times, and remember we had no calculators either.
http://www.johnkerl.org/doc/square-root.html
To find the square root of 345 without using tables, you can use a method called "estimation by division." Here's how you can do it:
1. Start by estimating the largest possible digit (let's call it x) that can be the first digit of the square root. To do this, you need to find the largest perfect square number that is less than or equal to 345. In this case, it is 19^2, which is 361. Therefore, x is 19 since it is the largest digit that can be the first digit of the square root of 345.
2. To find the next digit (let's call it y), bring down the next two digits of the given number, which is 45. Now, you need to find a value for y that satisfies the condition (20x + y) multiplied by y is less than or equal to 345.
- Begin by assuming y as the largest possible single-digit integer, which is 9. Substitute x=19 and y=9 into the equation: (20 × 19 + 9) × 9 = 3819, which is greater than 345.
- Reduce y by 1 and substitute it into the equation again until you find a value of y that satisfies the condition. You will eventually find that when y=6, (20 × 19 + 6) × 6 = 3366, which is less than 345.
3. Now, you have found the first two digits of the square root, which are 19. Repeat the process by bringing down the next two digits of the given number and finding the next digit of the square root.
4. Bring down the next two digits of the given number, which is 00. The remaining number is 156.
5. Double the value of the current root, which is 19. Let's call this value R.
- R = 2 × 19 = 38
6. Find the largest digit x that satisfies the condition (2R × x) multiplied by x is less than or equal to 156.
- Begin by assuming x as the largest possible single-digit integer, which is 9. Substitute R=38 and x=9 into the equation: (2 × 38 × 9) × 9 = 61452, which is greater than 156.
- Reduce x by 1 and substitute it into the equation again until you find a value of x that satisfies the condition. You will eventually find that when x=4, (2 × 38 × 4) × 4 = 2432, which is less than 156.
7. Now, you have found the first three digits of the square root, which are 19.4. Repeat the process by bringing down the next two digits of the given number and finding the next digit of the square root.
8. Bring down the next two digits of the given number, which is 00. The remaining number is 168.
9. Double the value of the current root, which is 19.4. Let's call this value R.
- R = 2 × 19.4 = 38.8
10. Find the largest digit x that satisfies the condition (2R × x) multiplied by x is less than or equal to 168.
- Begin by assuming x as the largest possible single-digit integer, which is 9. Substitute R=38.8 and x=9 into the equation: (2 × 38.8 × 9) × 9 = 62642.4, which is greater than 168.
- Reduce x by 1 and substitute it into the equation again until you find a value of x that satisfies the condition. You will eventually find that when x=8, (2 × 38.8 × 8) × 8 = 15703.04, which is less than 168.
11. Now, you have found the first four digits of the square root, which are 19.48. Repeat the process by bringing down the next two digits of the given number and finding the next digit of the square root.
Continue this process until you have the desired level of accuracy in the square root of 345.