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JadziaDax · 07-08-2003, 08:52 PM

How about 5 easy steps?

To find a cube root by the "longhand" method (without a calculator), we would proceed similarly to how we would find a square root by hand.

Step 1. Draw a cube root symbol, or radical, with the number whose root you are seeking underneath. Start with the decimal point and mark off digits in both directions in groups of three. Put a decimal point above the radical, and directly above the other decimal point.

Step 2. Start with the first group of 1, 2, or 3 digits. Find the largest cube of a single-digit integer less than it. Write the single digit above the radical, and its cube under the first group. Draw a line under that cube, and subtract it from the first group.

Step 3. Bring down the next group below the last line drawn. This forms the current remainder. Draw a vertical line to the left of the resulting number, and to the left of that line put three hundred times the square of the number above the radical, a plus sign, thirty times the number above the radical, a multiplication sign, an underscore character, another plus sign, another underscore character, the exponent 2, an equals sign, and some blank space for the answer.

Step 4. Pick the biggest digit D that would fit into both underscore places, and give a number such that D times it is less than the current remainder. Put it above the radical above the last group of digits brought down, and put it in each of the blanks where the underscore characters are. Compute the number given by the expression, and put it after the equals sign. Multiply D times that number, and put that below the current remainder, draw a horizontal line below that, and subtract, to give a new current remainder.

Step 5. If the current answer, above the radical, has the desired accuracy, stop. Otherwise, go back to step 3.

<center>OR</center>

An approximation method:

If you had to solve x^3 = 86, you could proceed by Newton's Method as follows:

f(x) = x^3 - 86 = 0
f'(x) = 3x^2

Then we can use an iterative formula to improve on the accuracy of the root. If x0 is our first guess at the root, then a better approximation is given by:

x1 = x0 - f(x0)/f'(x0)
x1 = x0 - (x0^3-86)/(3x0^2)

Now 4^3 = 64 and 5^3 = 125, so we know the root lies between 4 and 5, closer to 4. So let us start with x = 4.3

x1 = 4.3 - (4.3^3 - 86)/[3(4.3)^2] = 4.417
x2 = 4.414
x3 = 4.414005 and x4 = 4.414004962

If you use a calculator to work out cube root(86) you get the same value as x4, so the method gets the right answer quite quickly.

Hope either is helpful....

07-08-2003, 08:52 PM	#4 (permalink)
JadziaDax Loser Location: who the fuck cares?	How about 5 easy steps? To find a cube root by the "longhand" method (without a calculator), we would proceed similarly to how we would find a square root by hand. Step 1. Draw a cube root symbol, or radical, with the number whose root you are seeking underneath. Start with the decimal point and mark off digits in both directions in groups of three. Put a decimal point above the radical, and directly above the other decimal point. Step 2. Start with the first group of 1, 2, or 3 digits. Find the largest cube of a single-digit integer less than it. Write the single digit above the radical, and its cube under the first group. Draw a line under that cube, and subtract it from the first group. Step 3. Bring down the next group below the last line drawn. This forms the current remainder. Draw a vertical line to the left of the resulting number, and to the left of that line put three hundred times the square of the number above the radical, a plus sign, thirty times the number above the radical, a multiplication sign, an underscore character, another plus sign, another underscore character, the exponent 2, an equals sign, and some blank space for the answer. Step 4. Pick the biggest digit D that would fit into both underscore places, and give a number such that D times it is less than the current remainder. Put it above the radical above the last group of digits brought down, and put it in each of the blanks where the underscore characters are. Compute the number given by the expression, and put it after the equals sign. Multiply D times that number, and put that below the current remainder, draw a horizontal line below that, and subtract, to give a new current remainder. Step 5. If the current answer, above the radical, has the desired accuracy, stop. Otherwise, go back to step 3. <center>OR</center> An approximation method: If you had to solve x^3 = 86, you could proceed by Newton's Method as follows: f(x) = x^3 - 86 = 0 f'(x) = 3x^2 Then we can use an iterative formula to improve on the accuracy of the root. If x0 is our first guess at the root, then a better approximation is given by: x1 = x0 - f(x0)/f'(x0) x1 = x0 - (x0^3-86)/(3x0^2) Now 4^3 = 64 and 5^3 = 125, so we know the root lies between 4 and 5, closer to 4. So let us start with x = 4.3 x1 = 4.3 - (4.3^3 - 86)/[3(4.3)^2] = 4.417 x2 = 4.414 x3 = 4.414005 and x4 = 4.414004962 If you use a calculator to work out cube root(86) you get the same value as x4, so the method gets the right answer quite quickly. Hope either is helpful....