Square Root and Cubic Root

Content

What does it mean to get the root of a number?

Depending on the root index it is possible to obtain a number that is raised to the power of the radical, the number to be analyzed will result.

It is common to analyze the square root and the cube root of a number, but depending on the index it is the root that must be analyzed.

Note: Not representing a number in the index is considered as a square root.

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Root Parts

The root consists of 3 basic parts:

• Radicand (a): It is the number from which you want to know the root.
• Index (n): It is the value from which you want to get the root, the letter n represents the index.
• Root (r): It is the result of the operation.

ⁿ√a = r

The root of a number is proportional to raising the result of the root to the radial index will be equal to the radicand, which corresponds to the exponent or power part.

ⁿ√a = r ↔ r ⁿ = a

Learn more about: “Exponent”. →

Important: It is recommended to have reinforced knowledge for the exponent or potency of a number so that the root can be solved easily.

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Square root

The square root of a number is the number (root result) that squared exactly reproduces the given number (radical). It can be presented with index 2 or without index.

√a = ²√a

How to calculate the square root?

To obtain the square root of a number in some cases it is very simple to square that number. For example:

²√4 = ²√2² = 2
²√16 = ²√4² = 4

Of course, with practice it will be easy to obtain the square root of any number or an approximation.

Below is the procedure to obtain the square root, as an example the number 65536 is used.

√ 65536

1. You must separate in pairs from right to left the number from which you want to obtain the square root (The separation is represented by an apostrophe):
   
   √ 6 ´ 55 ´ 36
   


2. Calculate the square root of the first couple on the left and position the result on the root line, the corresponding subtraction is performed.
   
   
   √ 6 ´ 55 ´ 36   2    - 4                    2
   The corresponding number is 2 ² = 4, it could not be 3 ² = 9 since it is greater than 6.


3. The next pair of numbers is lowered and the number found in the root must be multiplied by "2".
   
   
   √ 6 ´ 55 ´ 36   2              - 4                  2 x 2 = 4   2    55
4. Now you must find a digit to add to the root line and do the corresponding multiplication, the result will be the subtraction.
   
   
   √ 6 ´ 55 ´ 36   2              - 4                  2 x 2 = 4   2    55          4    x    
   • 4.1. One method to find the digit is to take the number we know of n digits (In this case it corresponds to 4, therefore it is a digit), from the number 255 you must take n + 1 digits from left to right (Since n = 1, 1 + 1 = 2 digits should be considered), which corresponds to considering the number 25.
   
   
   • 4.2. Subsequently, the separation must be multiplied by 4 by a number that gives a result or approximately 25 (In number it must be an integer). The best way is to divide 25/4 and consider the whole part.
   
   
   • 4.3. The closest number to 25 is 6 x 4 = 24, but when doing the complete operation of 46 x 6 = 276 the result is greater than 255, therefore it is not possible to solve with the digit 6, the solution is very simple we go back a number now it would be 5 x 4 = 20 and when completing the operation 45 x 5 = 225, which allows to continue solving the root.
   
   
   The digit found must be placed in the root line.
   
   
   √ 6 ´ 55 ´ 36   2 5                - 4                  2 x 2 = 4   2    55          45 x 5 = 225
5. Perform the corresponding subtraction and if it is still possible to continue solving you should go back to step 3
   
   
   √ 6 ´ 55 ´ 36   2 5              - 4                  2 x 2 = 4   2    55          45 x 5 = 225 - 2    25   0    30

The above mentioned corresponds to the steps to obtain the square root of a number, if there is any doubt then the missing points are represented to find the square root of 65536 (consider that the steps are already repetitive).

• After subtracting, we lower the next number pair and the number at the root must be multiplied by "2".
  
  
  √ 6 ´ 55 ´ 36   2 5              - 4                  2 x 2 = 4   2    55          45 x 5 = 225 - 2    25          25 x 2 = 50   0    30    36
• Now you must find a digit to add to the line and do the corresponding multiplication.
  
  
  √ 6 ´ 55 ´ 36   2 5              - 4                  2 x 2 = 4   2    55          45 x 5 = 225 - 2    25          25 x 2 = 50   0    30   36    50    x    
  • Now n has the value of 2 digits (in this case it corresponds to 50), therefore the number 3036 must be taken n + 1 digits from left to right (Since n = 2, 2 + 1 = 3 must be considered) , which corresponds to consider 303.
  
  
  • Performing the corresponding operations we find that 50 x 6 = 300 therefore 506 x 6 = 3036.
  
  
• Finally place the digit found in the root line, perform the corresponding subtraction and observe that the residue is "0", therefore, the result of the root is 256
  
  
  √ 6 ´ 55 ´ 36   2 5 6              - 4                  2 x 2 = 4   2    55          45 x 5 = 225 - 2    25          25 x 2 = 50   0    30   36    506 x 6 = 3036       - 30   36                 0

To check if the root is correct we must square 256 in this case 256 squared is equal to 65536.

Square root with residue

In some cases the square root will have a residue. For example:

√ 1 ´ 34   1 1             - 1           1 x 2 = 2   0    34   21 x 1 = 21 -       21         13

To check the result, you must raise the result in the square root and add the residue, the result corresponds to the radicand.

11² = 121
121 + 13 = 134

To obtain a more exact value the decimal point is used, therefore, groups of zeros are added to the radicand after the decimal point and in the root result the decimal point must be considered.

Square root with decimals

It is the same method, only some observations should be considered.

The separation of couples is done to the left of the decimal point and another separation of couple is done to the right of the decimal point, for example:

You want to find the square root of 637.4

√ 637.4

First we must do the separation of the numbers by couple to the left of the number, therefore, the separation would be like this:

√ 6 ´ 37 . 4

Now the separation to the right of the decimal point, since you only have the number 4 would add a “0” since the separation must be in pairs.

Therefore, the square root of 637.4 with the corresponding separations would be:

√ 6 ´ 37 . 40

Whereas the point is a separator.

Since we have the separate root we must do the procedure to obtain the result, now all that would be missing would be to consider the point in the root result line.

√ 6 ´ 37 . 40   2 5.2             - 4                  2 x 2 = 4   2    37          45 x 5 = 225 - 2    25          25 x 2 = 50   0    12   40    502 x 2 = 1004    -    10   04             2   36

As can be seen in the previous procedure, there is still residue left, it is possible to continue adding zeros to the right of the point until the residue is “0” or depending on the number of decimals that are desired.

√ 6 ´ 37 . 40 00  2 5.2 4            - 4                      2 x 2 = 4   2    37              45 x 5 = 225 - 2    25              25 x 2 = 50   0    12   40       502 x 2 = 1004    -    10   04       252 x 2 = 504           2   36 00  5044 x 4 = 20176       -   2   01 76           0   34 24

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Cubic Root

The cube root of a number is the number (root result) that raised to the third power reproduces exactly the given number (radical).

With practice it will be easy to obtain the cube root of any number or an approximation.

³√27 = ³√3³ = 3
³√125 = ³√5³ = 5

How to calculate the cube root?

Below is the procedure to obtain the cube root, as an example the number 2299968 is used.

³√ 2299968

1. The number from which you want to obtain the cube root must be separated into the numbers three by three from right to left (The separation is represented by an apostrophe): ³√ 2 ' 299 ' 968
   
2. Calculate the cube root of the first couple on the left and position the result on the root line, the corresponding subtraction is performed.
   
   
   ³√ 2 ´ 299 ´ 968   1      - 1     1


3. The next pair of numbers is lowered and the number found in the root must be squared and multiplied by 3.
   
   
   ³√ 2 ´ 299 ´ 968   1      - 1                     3 x 1² = 3     1   299
4. Subsequently, the following digit of the root result must be found, therefore, the corresponding figure is divided by the previous result. Two digits from right to left must be removed, for example 1299 only 12 is considered.
   
   
   ³√ 2 ´ 299 ´ 968   1      - 1                     3 x 1² = 3     1   299             12 / 3 = 4
5. The result obtained is possibly the digit, for this we must check by performing the following steps:
   • 5.1. Perform a multiplication of: 3 that multiplies the root digits squared, by the found digit, by 100. 3 x 1² x 4 x 100 = 1200
   • 5.2. Perform a multiplication of: 3 that multiplies the digits of the root, by the digit found squared and by 10. 3 x 1 x 4² x 10 = 480
   • 5.3. Cube the found digit and multiply by 1. 4³ x 1 = 64
   • 5.4. When performing the previous 3 steps, it is only necessary to make the corresponding sum and verify if it is possible to subtract. 1200 + 480 + 64 = 1744

     Note: If it is not possible to subtract, then the value of the digit must be decreased by 1 and repeat the previous steps.

     Since doing the operation 1229 - 1744 would result in a negative number then the digit 4 is not correct.

     
     
     ³√ 2 ´ 299 ´ 968   1      - 1                     3 x 1² = 3     1   299             12 / 3 = 4 → 3
     
     

   Again the previous steps are done, now with the digit 3:

   3 x 1² x 3 x 100 = 900 3 x 1 x 3² x 10 = 270 3³ x 1 = 27 900 + 270 + 27 = 1197
   The digit found must be placed in the root result.
6. Perform the corresponding subtraction and if it is still possible to continue solving you should go back to step 3
   
   ³√ 2 ´ 299 ´ 968   1 3           - 1                     3 x 1² = 3     1   299             12 / 3 = 4 → 3   - 1  197     0   102
   
   

The above steps are the basics to get the cube root of a number, below is the complete operation to find the cube root of 2299968:

³√ 2 ´ 299 ´ 968   1 3 2              - 1                      3 x 1² = 3     1   299             12 / 3 = 4 → 3   - 1  197              3 x 13² = 507     0   102    968  1029 / 507 = 2.029 → 2        - 102   968                       0

Operation with digit 2:

3 x 13² x 2 x 100 = 101400
3 x 13 x 2² x 10 = 1560
2³ x 1 = 8
101400 + 1560 + 8 = 102968

As you can see the residue is 0, the result of the cube root is 132, to check only 132 must be raised to the cube, that means that 132 x 132 x 132 = 2299968.

Cubic root with residue

If the result of the cube root has a residue, it is checked in the same way and when the cube root result is added to the cube, the residue must be added, the result must be the number in the radicand.

Cubic root with decimal

The way to obtain is the same procedure of adding zeros, therefore, considering the separation of numbers to the left of the decimal point and the separation to the right of decimal point.

Important: In this case 3 zeros would be added.