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square roots
- Thread starter dfritzful
- Start date
How do I solve square roots?
Are you asking what the operation known as "square root" is/does? Or are you asking how you might be able to do them without a calculator? The question is unclear. We'll try to help if you can clarify what you mean.
Are you asking what the operation known as "square root" is/does? Or are you asking how you might be able to do them without a calculator? The question is unclear. We'll try to help if you can clarify what you mean.
How do i DO them without a calculator
Well the first thing I would do is to figure out whether the number is a perfect square or not. I don't have any trick to doing this per se except memorizing a bunch of them (e.g. 1-25) and using other math tricks to solve ones I haven't memorized.
After that take the number you're given and then see what perfect squares it falls between and then start squaring until you're close. Your problem set should usually come with an instruction for how many decimal places it needs to be rounded off to.
Does that help you any?
After that take the number you're given and then see what perfect squares it falls between and then start squaring until you're close. Your problem set should usually come with an instruction for how many decimal places it needs to be rounded off to.
Does that help you any?
How do I solve square roots?
The square root of a number N is the number "a" which, when multiplied by itself, results in the number N or N = axa. There is no formula for extracting the square root of a number. It can be obtained by means of the long division method or a simple estimation method.
Estimation Method
Newton's method of square root estimation often leads directly to the minimal solution.
1--Make an estimate of the square root of N = n.
2--Then, calculate n1 = [N/n + n]/2
3--For further accuracy, calculate n2 = [N/n1 + n1]/2
Example: Estimate the square root of 73.
n1 = [73/8 + 8]/2 = 8.5625
n2 = [73/8.5625 + 8.5625]/2 = 8.54402
The actual square root of 73 is 8.544003745.
The method gives a reasonably close estimate in two calculations.
<-------------------------------------------------------------------------------------------------->
Extracting the square root by one traditional division method is explained below.
What if you are seeking the square root of a two digit number.
The two digit number can be expressed in the form of a + b.
The square of a + b is a^2 + 2ab + b^2.
We now need a method of extracting the terms of the root a + b.
Clearly, the first term, a, is the square root of a^2 and the first term of the required root.
Subtracting a^2 from the given expression yields 2ab + b^2 or (2a + b)xb.
Hence, b, the second term of the root, will become the quotient when the remainder is divided by 2a + b.
The divisor consists of two terms: 1) The double of "a" and 2) the new term "b".
The division arrangement looks like this:
............a^2 + 2ab + b^2 (a + b
............a^2...........................
2a + b...........2ab + b^2
....................2ab + b^2
Example: Find the square root of 4x^2 - 24xy + 36y
.............4x^2 - 24xy + 36y (2x - 6y
.............4x^2...............................
4x - 6y...........-24xy + 36y
.....................-24xy + 36y
Lets examine the extraction of a numerical square root, say 8649.
By observation, 8649 lies between 8100 and 10,000 or between 90^2 and 100^2.
The square root consists of two numbers and lies between 90 and 100.
The first term of our square root is therefore 90.
Therefore:.........8649 (90 + 3................a^2 + 2ab + b^2
..................…....8100...........................a^2..................
180+3 = 183.......549.............2a + b.............2ab + b^2
.................…......549.............................…....2ab + b^2
Therefore, the square root of 8649 is 93
Square root of 74,529
By observation, 74,529 lies between 40,000 and 90,000 or 200^2 and 300^2.
The square root consists of two numbers and lies between 200 and 300.
The first term of our square is therefore 200.
Therefore:..................74,529 (200 + 70 + 3)
.........................…......40,000
400 + 70 = 470..........34,529
.....................….........32,900
540 + 3 = 543.............1,629
................…....….........1,629
Therefore, the square root of 74,529 is 273.
The following is a simplified procedure for deriving the answers.
Find the square root of 79,328.
Divide the number 79,328 into groups of two numbers from right to left.
7..93..29
What is the highest square less than 7? 4 of course, the square root of which is 2..
Thus, .......7..93..28 .00..00 ) 2 = the first digit of the answer.
................4............
................3...93 (bring down the next group of two integers)
Multiply the 2 by 2 and find the number to attach to the result that when multiplied is less than 393
Thus, .......7..93..28 .00..00) 2 8 = the second digit of the answer.
...............4............
...............3...93
48x8........3...84
.....................9..28 (bring down the next group of two integers)
Multiply the 28 by 2 and find the number to attach to the result that when multiplied is less than 929.
Thus, .......7..93..28 .00..00) 2 8 1 = the third digit of the answer.
...............4............
...............3...93
48x8........3...84
.....................9..28
561x1............5..61
.....................3..67..00
Multiply the 281 by 2 and find the number to attach to the result that when multiplied is less than 36,700.
Thus, .......7..93..28 .00..00) 2 8 1 .6 = the fourth digit of the answer.
...............4............
...............3...93
48x8........3...84
.....................9..28
561x1............5..61
.....................3..67..00
5626x6..........3..37..56
........................29..44..00
Multiply the 2826 by 2 and find the number to attach to the result that when multiplied is less than 294,400.
Thus, .......7..93..28 .00..00) 2 8 1 .6 5 = the fifth digit of the answer.
...............4............
...............3...93
48x8........3...84
.....................9..28
561x1............5..61
.....................3..67..00
5626x6..........3..37..56
........................29..44..00
56,325x5...........28..16..25
Thus, the square root of 79,328 to two decimals is 281.65...
Try some examples yourself.
The square root of a number N is the number "a" which, when multiplied by itself, results in the number N or N = axa. There is no formula for extracting the square root of a number. It can be obtained by means of the long division method or a simple estimation method.
Estimation Method
Newton's method of square root estimation often leads directly to the minimal solution.
1--Make an estimate of the square root of N = n.
2--Then, calculate n1 = [N/n + n]/2
3--For further accuracy, calculate n2 = [N/n1 + n1]/2
Example: Estimate the square root of 73.
n1 = [73/8 + 8]/2 = 8.5625
n2 = [73/8.5625 + 8.5625]/2 = 8.54402
The actual square root of 73 is 8.544003745.
The method gives a reasonably close estimate in two calculations.
<-------------------------------------------------------------------------------------------------->
Extracting the square root by one traditional division method is explained below.
What if you are seeking the square root of a two digit number.
The two digit number can be expressed in the form of a + b.
The square of a + b is a^2 + 2ab + b^2.
We now need a method of extracting the terms of the root a + b.
Clearly, the first term, a, is the square root of a^2 and the first term of the required root.
Subtracting a^2 from the given expression yields 2ab + b^2 or (2a + b)xb.
Hence, b, the second term of the root, will become the quotient when the remainder is divided by 2a + b.
The divisor consists of two terms: 1) The double of "a" and 2) the new term "b".
The division arrangement looks like this:
............a^2 + 2ab + b^2 (a + b
............a^2...........................
2a + b...........2ab + b^2
....................2ab + b^2
Example: Find the square root of 4x^2 - 24xy + 36y
.............4x^2 - 24xy + 36y (2x - 6y
.............4x^2...............................
4x - 6y...........-24xy + 36y
.....................-24xy + 36y
Lets examine the extraction of a numerical square root, say 8649.
By observation, 8649 lies between 8100 and 10,000 or between 90^2 and 100^2.
The square root consists of two numbers and lies between 90 and 100.
The first term of our square root is therefore 90.
Therefore:.........8649 (90 + 3................a^2 + 2ab + b^2
..................…....8100...........................a^2..................
180+3 = 183.......549.............2a + b.............2ab + b^2
.................…......549.............................…....2ab + b^2
Therefore, the square root of 8649 is 93
Square root of 74,529
By observation, 74,529 lies between 40,000 and 90,000 or 200^2 and 300^2.
The square root consists of two numbers and lies between 200 and 300.
The first term of our square is therefore 200.
Therefore:..................74,529 (200 + 70 + 3)
.........................…......40,000
400 + 70 = 470..........34,529
.....................….........32,900
540 + 3 = 543.............1,629
................…....….........1,629
Therefore, the square root of 74,529 is 273.
The following is a simplified procedure for deriving the answers.
Find the square root of 79,328.
Divide the number 79,328 into groups of two numbers from right to left.
7..93..29
What is the highest square less than 7? 4 of course, the square root of which is 2..
Thus, .......7..93..28 .00..00 ) 2 = the first digit of the answer.
................4............
................3...93 (bring down the next group of two integers)
Multiply the 2 by 2 and find the number to attach to the result that when multiplied is less than 393
Thus, .......7..93..28 .00..00) 2 8 = the second digit of the answer.
...............4............
...............3...93
48x8........3...84
.....................9..28 (bring down the next group of two integers)
Multiply the 28 by 2 and find the number to attach to the result that when multiplied is less than 929.
Thus, .......7..93..28 .00..00) 2 8 1 = the third digit of the answer.
...............4............
...............3...93
48x8........3...84
.....................9..28
561x1............5..61
.....................3..67..00
Multiply the 281 by 2 and find the number to attach to the result that when multiplied is less than 36,700.
Thus, .......7..93..28 .00..00) 2 8 1 .6 = the fourth digit of the answer.
...............4............
...............3...93
48x8........3...84
.....................9..28
561x1............5..61
.....................3..67..00
5626x6..........3..37..56
........................29..44..00
Multiply the 2826 by 2 and find the number to attach to the result that when multiplied is less than 294,400.
Thus, .......7..93..28 .00..00) 2 8 1 .6 5 = the fifth digit of the answer.
...............4............
...............3...93
48x8........3...84
.....................9..28
561x1............5..61
.....................3..67..00
5626x6..........3..37..56
........................29..44..00
56,325x5...........28..16..25
Thus, the square root of 79,328 to two decimals is 281.65...
Try some examples yourself.