How do you take the square root of 300?

Question

What is the correct methodology for factoring numbers without obvious roots/solutions? This is for the graduate record exam, so I need to know how to apply the correct methodology to more than one number.

Answer 1

Take two known square roots of similar values.
Extrapolate from the known answers.

Example: 20** = 400
16** = 256
300 is 44 above 256, 400 is 144 above 256
divide 44 by 144 = about .30
multiple .30 by (20 - 16) and get 1.2
add to 16 and get 17.2
the actual answer is 17.3 but this is an approximation

Answer 2

Oh, the obvious answer is to plug it into a calculator. You want to factor it first and make the answer general.
Divide by whatever you notice first and keep a list of every root.
300/3 = 100/5 = 20/5= 4/2=2
The roots are 5,5,3,2,2
You have the roots 5 and 2 twice so those are even square roots and 5X2 = 10. You have a 3 left over so
square root (300) = 10 X square root (3)
Easier and faster if you notice a perfect square root first.
300 = 3X100 and 100= 10^2 so
you get the same answer 10X square root (3)

Answer 3

Divide this up into factors, taking out the biggest perfect square you can. What is left will be one of the more common square roots you might know.

For this example √300 = √(100 * 3) = 10 √3 or 17.17

√180 = √(9 * 4 * 5) or 6 √5 or 13.42

Answer 4

√300 = √(100 x 3) = 10.√3

Answer 5

A general procedure?
Factor: 300 = 2*2*3*5*5
Collapse pairs outside the radical: 2*5*root(3)

Answer 6

sqrt(300)
= sqrt (100*3)
= 10sqrt3