Square Roots......An exercise in multiplication experimentation

As you know, the area of a square is found by multiplying its
length times its width.

Now since the length and the width are the same in a square, you are really just multiplying a number by itself  to get the area of a square.

That is why they refer to n^2 (n to the second power) as "n squared".

Now if we already KNOW the area of the square and we want to find the length of the sides, we are looking for the "SQUARE ROOT".

EX: Let's say the area of this square is 49 square centimeters.
How long is it on one side?  ........Well 7 x 7 = 49, so it must be 7 cm on each side.  We say that 7 is the SQUARE ROOT OF 49.

The funny black symbol is called the "radical sign" or square root symbol.

*NOTE: (Every positive number has 2 square roots, a positive one and a negative one.  When we are finding the sides of squares given their areas we only need the POSITIVE square root.)

You try these.  When you think you have the answer, run your mouse over the question to see if you are correct. 

here the area is 16 sq. in. (How long is the side?)

here the area is 81 sq. ft. (How long is the side?)

here the area is 1 yd^2 (How long is the side?)

Now the tricky part comes when you have a square whose area is NOT a perfect square number.

Let's say the square has an area of 39 square inches.  How long would it's side be?

Well it can't be 6, because 6x6 = 36 not 39.

It can't even be 7, because 7x7 = 49 not 39.

SO HERE IS WHERE WE START TO EXPERIMENT.

FIRE up your CALCULATOR and let's get started!

I think the square root will be closer to 6 so I will guess 6.3.  

Now we check it out.  Use your calculator to multiply 6.3x6.3 ,
you will get 39.69.  This is close but too big; I only need 39.

So now let's try 6.2 x 6.2 = 38.44....too small.

Let's try 6.24 x 6.24 ...... that one gives 38.9376.....WOW we are getting close to 39 but still a bit too small.

Let's try 6.25 x 6.25 ....... here we get 39.0625 ........just a hair too big.

How about 6.248 x 6.248 = ...........39.037504   ........still too big.

One more experiment, 6.247 x 6.247 = ........39.025009 

Keep experimenting, see if you can get it down to three zeros after the decimal point.

**In the old days, before calculators, this is the way people found square roots by hand.  They did tons of experimenting, and as they found the roots out to the thousandths place value, they created tables.  These tables were put in the back of math books for the students to use as reference.

Now calculators can give you really good estimates of square roots.   Some calculators can figure the values of square roots out to 10 decimal places.   BUT IT IS STILL AN ESTIMATE of the real square root!

Perfect square numbers like 1, 4, 9, 16, 25, 36, 49, 81, etc. have easy whole number square roots.  But if we try to find the square roots of non perfect squares we will always get non repeating, non terminating decimals. 

For example: SQR(8) is approximately equal to 2.828427125...

If you want to write the EXACT square root, you need to keep 8 under the radical symbol. This is EXACT notation.

The radical symbol acts like a set of parentheses when there is more than one number underneath it.  If you are asked to simplify the square root of 9 + 16, you must add the 9 and the 16 first to get 25.  Then take the SQR(25) which is +5, or -5.

"Squaring" a square root undoes the square root function.  
times = 8 

NOW TO SOLVE AN EQUATION:

Solve: Give the EXACT solution.

2 = d
d....13

Here you multiply along the diagonals since you have a proportion.

2 x 13 = d x d

26 = d²

^{To undo the square on the variable we take the SQUARE ROOT of both sides.}

SQR(26) = SQR(d²)

+/- SQR(26) = d   Always remember there are two square root solutions, the positive and the negative.