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I need help with square root integers
- Thread starter sammy56
- Start date
arthur ohlsten
Full Member
- Joined
- Feb 20, 2005
- Messages
- 852
square root 6^100
example
square root 2^4.
we know 2^4 is 16 and its square root is 4
square root of 2^4=[2^4]^1/2
when raised to a power you multiply
[2^4]^1/2 = 2^2 check
square root 6^100= [6^100]^1/2
square root 6^100= 6^50 answer
example
square root 2^4.
we know 2^4 is 16 and its square root is 4
square root of 2^4=[2^4]^1/2
when raised to a power you multiply
[2^4]^1/2 = 2^2 check
square root 6^100= [6^100]^1/2
square root 6^100= 6^50 answer
D
Deleted member 4993
Guest
arthur ohlsten said:square root 6^100
example
square root 2^4.
we know 2^4 is 16 and its square root is 4
square root of 2^4=[2^4]^1/2
when raised to a power you multiply
[2^4]^1/2 = 2^2 check
square root 6^100= [6^100]^1/2
square root 6^100= 6^50 answer <<< we started with -6
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 actural 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...
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 actural 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...
D
Deleted member 4993
Guest
The algorithm above is actually attributed to Babylonians - however, it "converges" to Newton-Raphson.TchrWill said: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 actural square root of 73 is 8.544003745.
The method gives a reasonably close estimate in two calculations.
arthur ohlsten
Full Member
- Joined
- Feb 20, 2005
- Messages
- 852
sorry about the -6
square root [-6]^100= [-6]^50
but -1 =i^2
[-6]^50=[i^2*6]^50
[i^2]^50 [6^50
i^100[6^50]
but
i^1=i
i^2=-1
i^3=-i
i^4=1
i^5=i
...
i^10=i
....
i^100=i
i^100[6^50]=i[6^50]
Arthur
please check math
s
square root [-6]^100= [-6]^50
but -1 =i^2
[-6]^50=[i^2*6]^50
[i^2]^50 [6^50
i^100[6^50]
but
i^1=i
i^2=-1
i^3=-i
i^4=1
i^5=i
...
i^10=i
....
i^100=i
i^100[6^50]=i[6^50]
Arthur
please check math
s
Sammy, what d'heck is your question anyway? :shock:sammy56 said:-6^100