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Thought-Process to Discover Knowledge

Welcome to nubtrek.

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.
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.

click on the content to continue..

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. 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" Procedure to finding Square Root of a number : Long division method is illustrated in the figure. slide-show version coming soon