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)`?
- `2 sqrt(2)`
- `2 sqrt(2)`
The answer is "`2 sqrt(2)`".
The prime factorization is given as
The prime factorization method is suitable for square roots resulting in integer values.
Square root of a number can be given as
`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`.
`x^2 color(coral)(- a^2 text( at 100s place)) = y`
`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.
`=(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
`=color(coral)(2) text( at 10s place ) + color(deepskyblue)(3) text( at units place)`
This procedure is illustrated in the figure. 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)`?
The answer is "`263`"
Procedure to finding Square Root of a number : Long division method is illustrated in the figure.