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How to Calculate Cube Root??

Faiz Ahmed ( Article Trainee) (1731 Points)

01 September 2010

How to calculate cube root on normal calculator having basic functions(allowed in exam hall)??

Thnx in advance!!

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7 Replies

sivaram (Asst Mgr-Taxation) (6918 Points)
Replied 01 September 2010

I think we can use scientific calculators in exams so that you can use it for cube root if not allowed then definitely value will be supplied in the question itself

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Faiz Ahmed ( Article Trainee) (1731 Points)
Replied 01 September 2010

Scientific calculators are not allowed in exams. It will be better if we know how to calculate on basic calculator. I have forgotten how to calculate it. Its easy but i am not able to recall. If they don't give value , whole problem will become another big problem in exams. Its better to know it in advance than to take risk.

Manmohan ACA, CS (Chartered Accountant ) (14243 Points)
Replied 01 September 2010

Cube root

(Without calculator)

( using algorithm)

Definition: This describes a "long hand" or manual method of calculating or extracting cube roots. Calculation of a cube root by hand is similar to long-hand division or manual square 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.

source : file submitted by me

Faiz Ahmed ( Article Trainee) (1731 Points)
Replied 01 September 2010

Dear Manmohan,

I appreciate your reply. This seems to be a very long process . In exams we can't afford to spend so much time. If calculations become wrong, whole thing will create irritation.

I would be highly thankful to you if you can provide me with some short cut(not time consuming).

(Guest)

1) write the number on your calculator

2) press the square root button 12 times

3) subtract 1

4) divide by n where n is the nth root. For example if you are finding the cube root of a number, n will be 3.

5) add 1

6) press "multiply button and then equal to button" 12 times i.e. multiplty equal to multiply equal to .....

5 Like

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Sunshine (Helping All) (10575 Points)
Replied 29 October 2010

hit and trial...best hai....koi fikar nai.......

1 Like

pavithra doss (student) (234 Points)
Replied 29 October 2010

Originally posted by : Prerna
	1) write the number on your calculator 2) press the square root button 12 times 3) subtract 1 4) divide by n where n is the nth root. For example if you are finding the cube root of a number, n will be 3. 5) add 1 6) press "multiply button and then equal to button" 12 times i.e. multiplty equal to multiply equal to .....