How do you find the square root of a number without a calculator

I am going to be using the Pythagorean Theorem soon and I know all the steps until I get to where you find the square root of the number. I would rather do it all by hand, not with a calculator. The only videos i have found just said about using a factor tree to get the answer like "seven times the square root of two". But when you are using the Pythagorean Theorem you need an actual number, not another equation.

The only way, without a calculator, is the Estimate and Multiply method.

You first ESTIMATE the aquare root, then multiply it by itself and see how close you get.

What is the Square root of 3?

3 = 1.5 * 2 1.4^2 = 2.25 2^2 = 4
The square root of 3 is between 1.5 and 2.
ESTIMATE 1.7 ----- 1.7 * 1.7 = 2.89
ESTIMATE 1.8 ----- 1.8 * 1.8 = 3.24
ESTIMATE 1.75 ----- 1.75 * 1.75 = 3.0625
ESTIMATE 1.74 ----- 1.74 * 1.74 = 3.0276
ESTIMATE 1.72 ----- 1.72 * 1.7 = 2.9584


And so on, until you get close enough.

Are we talking about a number that will come out nicely? Like sqr root of 49? or something not nice like the square root of 2?


If were talking about nice numbers..just do the multiplication and eventually you'll get it right.

ex.
196

12x12=144..nope
13x13=169
14x14=196..e z!

Newton's Method of successive approximation.

Or using tables, which we used before calculators existed. Or a slide rule, to 2 places of decimal.

See the algorithm method at the link below.

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for pythagorean theorum, you don't need to do this, just leave it in its radical form