Calculate Square Root . . . without using a square root calculator
Example
of
Direct Calculation
Precise, digit by digit method used to Calculate Square Roots (The Chinese Method).
Video demonstrating this method  click here
Calculate Square Root:
Step 1:
Beginning at the decimal point, divide the radicand into groups of two digits in both directions. The decimal can be extended as far as you wish.
Step 2:
Beginning on the left, select the first group of one or two digits.
In this example, the first group of digits in the radicand is simply the number 2.
Step 3:
If possible, find the perfect square root (expressed as an integer) of the first group of digits in the radicand.
Or, If the first group of digits does not have a perfect square root (expressed as an integer), find the perfect square root of the number closest to, but less than, the first group of digits.
In this example, the number 2 does not have a perfect square root (which can be expressed as an integer).
The number closest to, but less than, the number 2 which has a perfect square root is the number 1.
The perfect square root of the number 1 is also1. Write this number above the radical sign as shown below.
Step 4:
Square the answer to step 3, and subtract it from the first group of numbers.
In this example, the number 1 squared is subtracted from the number 2.
Step 5:

A  Bring down the next group of two digits.
In this example, bring down 73.

B  Double the result of step 3 (shown as a slate color), and write on lower left.
In this example, the number 1 x 2 = 2.

C  Add another digit to the number computed in "B" (the slate colored number) on the lower left. Choose the new digit so that the new number created by the additional digit can be divided into the number on the immediate right the same number of times as the new digit.
In this example, the new digit is 6. The resulting number (26), can be divided into 173 a total of 6 times.
Step 6:
Repeat steps A, B, and C shown in step 5.

A  Bring down the next group of two digits.
In this example, bring down 00.

B  Double the result of step 5 (shown as a slate color), and write on lower left.
In this example, the number 16 x 2 = 32.

C  Add another digit to the number computed in "B" (the slate colored number) on the lower left. Choose the new digit so that the new number created by the additional digit can be divided into the number on the immediate right the same number of times as the new digit.
In this example, the new digit is 5. The resulting number (325), can be divided into 1700 a total of 5 times.
Step 7:
Repeat steps A, B, and C shown in step 5.

A  Bring down the next group of two digits.
In this example, bring down 00.

B  Double the result of step 6 (shown as a slate color), and write on lower left.
In this example, the number 165 x 2 = 330.

C  Add another digit to the number computed in "B" (the slate colored number) on the lower left. Choose the new digit so that the new number created by the additional digit can be divided into the number on the immediate right the same number of times as the new digit.
In this example, the new digit is 2. The resulting number (3302), can be divided into 7500 a total of 2 times.
Step 8:
Repeat steps A, B, and C shown in step 5.

A  Bring down the next group of two digits.
In this example, bring down 00.

B  Double the result of step 7 (shown as a slate color), and write on lower left.
In this example, the number 1652 x 2 = 3304.

C  Add another digit to the number computed in "B" (the slate colored number) on the lower left. Choose the new digit so that the new number created by the additional digit can be divided into the number on the immediate right the same number of times as the new digit.
In this example, the new digit is 2. The resulting number (33042), can be divided into 89600 a total of 2 times.
The final answer is:
to 3 decimal points
(not rounded)
Cube Roots: calculating cube roots by hand is more complicated than computing square roots by hand. For information on cube roots, see the following web sites:
http://www.itl.nist.gov/div897/sqg/dads/HTML/cubeRoot.html
http://mathforum.org/library/drmath/view/52605.html
http://www.mathpath.org/Algor/cuberoot/algor.cube.root.htm
Example
of
Newton's Method
NewtonRaphson method: consecutive numerical approximations used to
Calculate Square Roots (approximations become more and more accurate
with each iteration).
The three steps used to Calculate Square Roots are:

A  Initial Estimate of the square root of the radicand. (The initial estimate can be any number. Subsequent steps will refine the estimate.)

B  Divide the radicand by the estimate in part A.

C  refine the estimate by computing the Average of the estimate (from part A) and the quotient (from part B).
Video demonstrating this method  click here
Calculate Square Root:
1st Approximation: Calculate Square Root  Newton's Method

A  Initial Estimate
of the square root of the radicand.
In this example, a good initial estimate for the square root of 273 is 15
(However, the initial estimate could be "any" number.)

B  Divide the radicand by the estimate in part A.
In this example, divide 273 by 15.

C  1st Approximation  refine the initial estimate by computing the Average of the estimate (from part A) and the quotient (from part B).
In this example, find the average of 15 and 18.2.
1st Approximation of Square Root is:
2nd Approximation: Calculate Square Root  Newton's Method

A  Initial Estimate for the 2nd Approximation of the square root of the radicand. Use the results of the 1st Approximation, part C, above.
In this example, the initial estimate will be 16.6

B  Divide the radicand by the estimate in part A.
In this example, divide 273 by 16.6.

C  2nd Approximation  refine the initial estimate by computing the Average of the estimate (from part A) and the quotient (from part B).
In this example, find the average of 16.6 and 16.445783.
2nd Approximation of Square Root is:
3rd Approximation: Calculate Square Root  Newton's Method

A  Initial Estimate for the 3rd Approximation of the square root of the radicand. Use the results of the 2nd Approximation, part C, above.
In this example, the initial estimate will be 16.522892

B  Divide the radicand by the estimate in part A.
In this example, divide 273 by 16.522892.

C  3rd Approximation  refine the initial estimate by computing the Average of the estimate (from part A) and the quotient (from part B).
In this example, find the average of 16.522892 and 16.522531.
3rd Approximation of Square Root is:
(Final answer for this example)
Guess & Check Method
The Guess & Check Method used to Calculate Square Roots is exactly what the
name implies: Guess what the value of the square root might be, then multiply
your guess by itself to see if the product is equal to the original radicand.
The three steps used to Calculate Square Roots are:

A  Guess  Estimate the square root of the radicand. (The initial estimate can be any number. Subsequent steps will refine the estimate.)

B  Multiply the estimate by itself.

C  Check  Compare the product obtained in step "B" (above) with the radicand.

D  Refine the Estimate . If the product obtained in step "B" (above) is greater than the radicand, decrease the estimate. If the product obtained in step "B" (above) is less than the radicand, increase the estimate.

E  Repeat steps B through D until the estimate is an accurate square root of the radicand.
Calculate Square Root:
1st Approximation: Calculate Square Root  Guess & Check Method

A  Guess 
Initial Estimate of the square root of the radicand.
In this example, a good initial estimate for the square root of 273 is 15
(However, the initial estimate could be "any" number.)

B  Multiply the estimate by itself.
In this example, multiply 15 by 15.

C  Check  Compare the product obtained in step "B" (above) with the radicand.
In this example, 225 is less than 273.

D  Refine the Estimate . If the product obtained in step "B" (above) is greater than the radicand, decrease the estimate. If the product obtained in step "B" (above) is less than the radicand, increase the estimate.
In this example, since 225 is less than 273 the estimate will be increased from 15 to 16.
Refined estimate of Square Root is:
2nd Approximation: Calculate Square Root  Guess & Check Method

A  Guess 
Use the results of the 1st Approximation, part D, above..
In this example, the estimate for the square root of 273 is raised to 16

B  Multiply the estimate by itself.
In this example, multiply 16 by 16.

C  Check  Compare the product obtained in step "B" (above) with the radicand.
In this example, 256 is still less than 273.

D  Refine the Estimate . If the product obtained in step "B" (above) is greater than the radicand, decrease the estimate. If the product obtained in step "B" (above) is less than the radicand, increase the estimate.
In this example, since 256 is less than 273 (but closer than the original estimate) the estimate will be increased a small amount, from 16 to 16.5.
Refined estimate of Square Root is:
3rd Approximation: Calculate Square Root  Guess & Check Method

A  Guess 
Use the results of the 1st Approximation, part D, above..
In this example, the estimate for the square root of 273 is raised to 16.5

B  Multiply the estimate by itself.
In this example, multiply 16.5 by 16.5.

C  Check  Compare the product obtained in step "B" (above) with the radicand.
In this example, 272.25 is still less than 273, but it is much closer than the other estimates.

D  Refine the Estimate . If the product obtained in step "B" (above) is greater than the radicand, decrease the estimate. If the product obtained in step "B" (above) is less than the radicand, increase the estimate.
In this example, since 272.25 is less than 273 (but much closer than the other two estimates) the estimate should be increased slightly.
Refined estimate of Square Root: these steps must be
repeated over and over again until the degree of accuracy
required is achieved.