From: Everhart [everhart@gce.com] Sent: Thursday, April 26, 2001 7:47 PM To: gce at work Subject: [Fwd: Formula] -------- Original Message -------- Subject: Formula Date: Thu, 26 Apr 2001 15:04:30 -0400 From: Everhart Reply-To: Everhart@GCE.Com To: george sharp Use a fixed-width font to view this answer. To find a cube root by the "longhand" method, we proceed very much as we do to find a square root by hand. I intersperse numbered steps with an example. We will find the cube root of 113 to two decimal places. 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. . 3/----------- \/ 113.000 000 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. 4. 3/----------- \/ 113.000 000 64 ------- 49 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. 4. 3/----------- \/ 113.000 000 64 ------- 4800+120*_+_^2=???? | 49 000 4. Pick the biggest digit D which 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. 4. 8 3/----------- \/ 113.000 000 64 ------- 4800+120*8+8^2=5824 | 49 000 46 592 ---------- 2 408 5. If the current answer, above the radical, has the desired accuracy, stop. Otherwise, go back to step 3. Step 3: 4. 8 3/----------- \/ 113.000 000 64 ------- 4800+120*8+8^2=5824 | 49 000 46 592 ---------- 691200+1440*_+_^2=?????? | 2 408 000 Step 4: 4 . 8 3 3/----------- \/ 113.000 000 64 ------- 4800+120*8+8^2=5824 | 49 000 46 592 ---------- 691200+1440*3+3^2=695529 | 2 408 000 2 086 587 --------- 321 413 Step 5: Stop. Thus the cube root of 113 to two decimal places is 4.83. Checking, 4.83^3 = 112.6786, and 4.84^3 = 113.3799, so the answer is correct. -Doctor Rob, The Math Forum