Approximate square root of, 65, end square root
65
65
by following the steps below.
square root of, 65, end square root
65
65
must lie between the whole numbers
and
8 and 9.
To approximate the square root of 65, we can use the method of averaging.
1. Start by taking the average of the numbers 8 and 9:
(8 + 9) / 2 = 8.5
2. Square 8.5 to see if it is closer to 65:
8.5 * 8.5 = 72.25
3. Since 72.25 is greater than 65, we know that the square root of 65 will be closer to 8.
4. Repeat the process, but now take the average of 8 and 8.5:
(8 + 8.5) / 2 = 8.25
5. Square 8.25 to see if it is closer to 65:
8.25 * 8.25 = 68.0625
6. Again, 68.0625 is greater than 65, so we know that the square root of 65 will be even closer to 8.
By repeating this process multiple times, we can get a more accurate approximation. However, for simplicity, let's stop here and conclude that the approximate square root of 65 is 8.
Approximate square root of, 65, end square root
65
65
by following the steps below.
square root of, 65, end square root
65
65
must lie between the whole numbers
8
8 and
9
9 because 8, squared, equals, start box, 64, end box
8
2
=
64
8
2
=
64
and 9, squared, equals, start box, 81, end box
9
2
=
81
9
2
=
81
, and 65 lies between these values.
Drag square root of, 65, end square root
65
65
based on your estimate above:
Real
√65
__
_
The estimate obtained above is 8.
Now let's calculate the square root of 65 using this estimate.
To find a closer approximation, we can use the Newton's method of approximation.
1. Start with the estimate 8.
2. Use the formula:
next approximation = (previous approximation + (65 / previous approximation)) / 2
next approximation = (8 + (65 / 8)) / 2
next approximation = (8 + 8.125) / 2
next approximation = 16.125 / 2
next approximation = 8.0625
By using this formula, the next approximation is 8.0625.
3. Repeat this process by plugging the new approximation back into the formula:
next approximation = (8.0625 + (65 / 8.0625)) / 2
next approximation = (8.0625 + 8.0517) / 2
next approximation = 16.1142 / 2
next approximation = 8.0571
4. Continue iterating this process until the desired level of accuracy is achieved. However, let's stop here and conclude that the approximate square root of 65 is 8.0571.
Approximate square root of, 65, end square root
65
65
by following the steps below.
square root of, 65, end square root
65
65
must lie between the whole numbers
8
8 and
9
9 because 8, squared, equals, start box, 64, end box
8
2
=
64
8
2
=
64
and 9, squared, equals, start box, 81, end box
9
2
=
81
9
2
=
81
, and 65 lies between these values.
Drag square root of, 65, end square root
65
65
based on your estimate above:
Real
√65
__
_
To one decimal place, square root of, 65, end square root
65
65
must lie between
and
Based on the estimate of 8 obtained above, we can now refine it to one decimal place.
To do that, we can use the same method of averaging as before.
1. Start with the estimate 8.
2. Take the average of 8 and 9:
(8 + 9) / 2 = 8.5
3. Square 8.5 to see if it is closer to 65:
8.5 * 8.5 = 72.25
4. Since 72.25 is greater than 65, we know that the square root of 65 will be closer to 8.5 than 9.
Now we can conclude that the approximate square root of 65, to one decimal place, lies between 8 and 8.5.
To approximate the square root of 65, we can follow these steps:
Step 1: Start by finding the two perfect square numbers around 65. The perfect square numbers that are smaller and larger than 65 are 64 (8^2) and 81 (9^2).
Step 2: Take the square root of both those numbers:
Square root of 64 is 8.
Square root of 81 is 9.
Step 3: Since the square root of 65 must fall between 8 and 9, we can approximate it to a decimal by subtracting them:
9 - 8 = 1.
Step 4: Divide the number 65 by both 8 and 9 and find the difference between the actual and the estimated square root. This will help us approximate the square root further.
When 65 is divided by 8, we get 8.125. The difference between the actual square root of 65 (approximately 8.0622577) and the estimated square root in Step 3 (1) is 8.0622577 - 8.125 = -0.0622577.
When 65 is divided by 9, we get 7.222222. The difference between the actual square root of 65 (approximately 8.0622577) and the estimated square root in Step 3 (1) is 8.0622577 - 7.222222 = 0.8400357.
Step 5: Compare the differences in Step 4 and choose the more accurate approximation. In this case, 0.8400357 is closer to 0 than -0.0622577, so the more accurate approximation is 9 - 0 (since the difference is close to 0) = 9.
Therefore, the approximate square root of 65 is 9.
To approximate the square root of 65, you can follow these steps:
1. Identify the two whole numbers between which the square root of 65 lies. In this case, we are looking for the whole numbers between which the square root of 65 lies. We know that the square root of 65 is greater than the square root of 64 (which is 8) and less than the square root of 81 (which is 9). Therefore, the square root of 65 lies between 8 and 9.
So, the approximate square root of 65 lies between 8 and 9.