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Thought-Process to Discover Knowledge

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Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.continue
mathsExponentsSquares and Square Roots

Finding Square Root: Long Division method

In this page, finding square root using long division method, is introduced in a simple thought process.


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What is the `sqrt(8)`?

  • `4`
  • `2 sqrt(2)`
  • `2 sqrt(2)`

The answer is "`2 sqrt(2)`".

The prime factorization is given as
`sqrt(8)`
`=sqrt(2xx2xx2)`
`=2xxsqrt(2)`

The prime factorization method is suitable for square roots resulting in integer values.

Square root of a number can be given as
`x^2=(10a+b)^2`
`x^2=100a^2+(20ab)+b^2`

Note:
`b^2` is a `2` digit number with tens-units places
`100a^2` is a number that has `00` at tens-units places

This understanding gives a method to eliminate `b` and look at `a` to choose the highest digit of the square root.

`x^2=color(coral)(100a^2)+color(deepskyblue)((20a+b) xx b)`
`x^2 color(coral)(- 100a^2)= color(deepskyblue)((20a+b) xx b)`

The rearranged one with `(20a+b)` gives the method to multiply `a` by `2` (which is `2a`) and append a value `b`, which is `20a+b`. Then multiply `b` to `20a+b`.

step 1
`x^2 color(coral)(- a^2 text( at 100s place)) = y`
step 2
`y - color(deepskyblue)((2a text( joined with ) b) xx b)`

In this process, the choice of `a` and `b` make the square root `x=10a+b = a text( joined with )b`

The above process is explained for 2 digit square root and is easily extended for higher number of digits.

`529`
`=23^2`
`=(10xx2+3)^2`
`=(10xx2)^2+ 2 xx 10xx2 xx 3+3^2`
`=color(coral)(100xx2^2)+color(deepskyblue)((2xx2xx10 + 3)xx3)`
`=color(coral)(2^2 text( 100s place ))+color(deepskyblue)((2xx2 text( joined )3 = 43)xx 3)`

The above steps is used in reverse when the square root is not known
`sqrt(529)`
`=color(coral)(2) text( at 10s place ) + color(deepskyblue)(3) text( at units place)`
`=23`

This procedure is illustrated in the figure. square root by long division method The number is split as `5,29`.

Consider `5` first and choose `2` as the first digit.
`5-2xx2 = 1`
The first step is completed with `2`.

Then `129` is considered and the first digit `2` multiplied `2` is `4`. This `4` is the tens position and second digit `3` is chosen.
`43 xx 3 = 129`.

The square root of `529` is `23`.

Solved Exercise Problem:

What is `sqrt(69169)`?

  • `263`
  • `263`
  • `11`

The answer is "`263`" square root long division

Procedure to finding Square Root of a number : Long division method is illustrated in the figure.
square root long division

                            
slide-show version coming soon