How To Manually Find A Square Root And A Cube Root

How to manually find a square root and a cube root

CUBE ROOT Suppose you need to find the cube root of 55,742,968. Set up a "division" with the number under the radical. Mark off triples of digits, starting from the decimal point and working left. (The decimal point is a period (.), and commas (,) mark triples of digits.) ____________ \/ 55,742,968. Look at the leftmost digit(s) (55 in this case). What is the largest number whose cube is less than or equal to it? It is 3, whose cube is 27. Write 3 above, write the cube below and subtract. __3_________ \/ 55,742,968. -27 ---28 Now bring down the next three digits (742). __3_________ \/ 55,742,968. -27 ---28742 Coming up with the next "divisor" is more involved than for square roots. First bring down 3 times the square of the number on top (3 × 3²=27) leaving room for two more digits (27_ _). __3_________ \/ 55,742,968. -27 ---27_ _) 28742 What is the largest number that we can put in the next position and multiply times the divisor and still be less than or equal to what we have? (Algebraically, what is d such that d × 2700 ≤ 28742?) 10 might work (since 10 × 2700 = 27000), but we can only use a single digit, so we'll try 9.

__3___9_____ \/ 55,742,968. -27 ---27_ _) 28742

The second step in making the divisor is adding 3 times the previous number on top (3) times the last digit (9) times 10 (3 × 3 × 9 = 81 × 10 = 810) and the square of the last digit (9² = 81). 2700 810 + 81 ----3591 Our new divisor is 3591. __3___9_____ \/ 55,742,968. -27 ---3591) 28742 Multiply by the last digit (9 × 3591 = 32319) and subtract. But that is too big! So we'll try 8 as the next digit instead. __3___8_____ \/ 55,742,968. -27 ---27_ _) 28742 We repeat the second step of adding 3 times the previous number on top (3) times the last digit (8) times 10 (3 × 3 × 8 = 72 × 10 = 720) and the square of the last digit (8² = 64). 2700 720 + 64 ----3484 Our new divisor is 3484. __3___8_____ \/ 55,742,968. -27 ---3484) 28742 Now multiply by the last digit (8 × 3484 = 27872) and subtract. __3___8_____ \/ 55,742,968.

-27 ---3484) 28742 -27872 -----870

We are ready to start over on the next digit. Bring down the next three digits. The divisor starts as 3 times the square of the number on top (3 × 38²=4332) leaving room for two more digits (4332_ _). __3___8_____ \/ 55,742,968. -27 ---3484) 28742 -27872 -----4332_ _) 870968 It looks like 2 is the next digit. __3___8___2_ \/ 55,742,968. -27 ---3484) 28742 -27872 -----4332_ _) 870968 Add 3 times the previous number on top (38) times the last digit (2) times 10 (3 × 38 × 2 = 228 × 10 = 2280) and the square of the last digit (2² = 4). 433200 2280 + 4 ------435484 Our new divisor is 435484. __3___8___2_ \/ 55,742,968. -27 ---3484) 28742 -27872 -----435484 ) 870968 Now multiply by the last digit (2 × 435484 = 870968) and subtract. __3___8___2_ \/ 55,742,968. -27 ----

3484) 28742 -27872 -----435484 ) 870968 -870968 ------0 So the cube root of 55742968 is 382. You can continue to get as many decimal places as you need: just bring down more triples of zeros. Why does this work? Consider (10A + B)³ = 1000A³ + 3 × 100A²B + 3 × 10AB² + B³ and think about finding the volume of a cube. The volume of the three thin plates is 3 × 100A²B. The volume of the three skinny sticks is 3 × 10AB². The tiny cube is B³. If we know A and the volume of the cube, S, what B should we choose? We previously subtracted A³ from S. To scale to 1000A³, we bring down three more digits (a factor of 1000) of the length of S. We write down 3 times A squared (3A²), but shifted two places (100 × 3A² or 3 × 100A²). We estimate B. We add 30 times A times B (30 × AB or 3 × 10AB) and B squared. Multiplying that by B gives us 3 × 100A²B + 3 × 10AB² + B³. When we subtract that from the remainder (remember we already subtracted A³), we have subtracted exactly (10A + B)³. That is, we have improved our knowledge of the cube root by one digit, B. We take whatever remains, scale again by 1000, by bringing down three more digits, and repeat the process. SQUARE ROOT Suppose you need to find the square root of 66564. Set up a "division" with the number under the radical. Mark off pairs of digits, starting from the decimal point and working left. (Here the decimal point is a period (.) and commas (,) mark pairs of digits.)

___________ \/ 6,65,64. Look at the leftmost digit(s) (6 in this case). What is the largest number whose square is less than or equal to it? It is 2, whose square is 4. Write 2 above, write the square below and subtract. __2________ \/ 6,65,64. -4 ---2 Now bring down the next two digits (65). The next "divisor" is double the number on top (2x2=4) and some other digit in the units position (4_). __2________ \/ 6,65,64. -4 ----4_ ) 265 What is the largest number that we can put in the units and multiply times the divisor and still be less than or equal to what we have? (Algebraically, what is d such that d × 4d ≤ 265?) It looks like 6 might work (since 6 * 40 = 240), but 6 is too big, since 6 * 46 = 276. __2__6_____ \/ 6,65,64. -4 ----46 ) 265 276

How to manually find a square root and a cube root -225

------40 Repeat: bring down the next two digits, and double the number on top (2x25=50) to make a "divisor", with another unit. __2__5_____ \/ 6,65,64. -4 ----45 ) 265 -225 ------50_ ) 4064 It looks like 8 would work. Let's see. __2__5__8__ \/ 6,65,64. -4 ----45 ) 265

Why does this work?

-225 ------508 ) 4064 -4064 ------

TOO BIG

0 So the square root of 66564 is 258. You can continue for as many decimal places as you need: just bring down more pairs of zeros.

So try 5 instead. __2__5_____ \/ 6,65,64. -4 ----45 ) 265

__1__6.8_4_0_4_2_7_5_... 2,83.6 -1 ----26 ) 183 -156 -----328 ) 2760 -2624 ------3364 ) 13600 -13456 -------33680 ) 14400 -0 --------336804 ) 1440000 -1347216 ---------3368082 ) 9278400 -6736164 ----------33680847 ) 254223600 -235765929 -----------336808545 ) 1845767100 -1684042725 ----------161724375 \/

Here is an example spanning the decimal point. When a number does not have a rational square root, you can continue calculating (significant) digits as long as you wish.

Consider (10A + B)² = 100A² + 2 × 10AB + B² and think about finding the area of a square. Remember that 10A + B is just the numeral with B in the units place and A in the higher position. For 42, A=4 and B=2, so 10 × 4 + 2 = 42.

The area of the two skinny rectangles is 2 × 10A × B. The tiny square is B². If we know A and the area of the square, S, what B should we choose? We previously subtracted A² from S. To scale to 100A², we bring down two more digits (a factor of 100) of the size of S. We write down twice A (2A), but shifted one place to leave room for B (10 × 2A or 2 × 10A). Now we add B to get 2 × 10A + B. Multiplying by B gives us 2 × 10AB + B². When we subtract that from the remainder (remember we already subtracted A²), we have subtracted exactly (10A + B)². That is, we have improved our knowledge of the square root by one digit, B. We take whatever remains, scale again by 100, by bringing down two more digits, and repeat the process.