More Square Roots

Ok...you have learned one method of approximating square roots.  Now, often we have a table of square roots up to 100 to use, so we often don't have to do this.  But, what if we have something like:

  ___
√5.3

and we want to approximate it as closely as we can according to a square root table.  We can see it lies between 2 and 3 (2² = 4 and 3² = 9), and that it is closer to 2, but it would be hard to estimate it.  We also know it lies between √6 and √5, about .3 of the distance between √6 and √5.  You can always use a calculator, but there is also a method called interpolation.

                                         __
Ok...we can assume that √5.3 is about √5 + .3(√6 - √5)

Using a square root table we get:

2.236 + .3(2.449 - 2.236)

2.236 + .3(.213)

2.236 + .0639 = 2.299

One could probably round this to 2.3, since it is very close.

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Alright, now we have down the basics.  When working with radicals, especially square roots since they turn up so often, we will sometimes leave them in terms of radicals.  This means we don't find the exact value for the square root.  This is useful because it saves time and effort when you are working with them, because sometimes you may find they cancel out.

However, if you do this, you should know how to simplify radicals.  One thing you usually must do, is if the radical appears in the denominator of a fraction, move it to the numerator.  To do this, we use the property that √a * √a = a

Let's take an example...   9
                                      √3

We must make an equivalent fraction with the radical in the numerator...so, we multiply both the numerator and denominator by √3 (this is like multiplying by one, but it is written as a number over itself). Also, b*√a can be written without the * sign.

         _        _         _
 9 * √3 = 9√3 = 3√3
√3* √3      3

Let's do another one.  In this one, we will use a variable and assume we don't know it's value, and only want to simplify it.

          _        _      _
 x  * √x = x√x = √x
√x * √x      x

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Now for another kind of simplifying radicals.  For example √72 is not simplified.  This is because there are perfect square factors of the radicand (number inside the radical sign).  To simplify it, we find the largest perfect square factor of the radicand, factor it out of the radicand (divide the radicand by it), and multiply the remaining square root by it's square root.

Yeah, that was kind of a weird explanation...maybe it would be easier to see with an example...
  __
√72...the largest perfect square factor of the radicand is 36.  So, we divide 72 by 36 and get 2...then we multiply what we now have, √2 by √36 which is 6...
    _
6√2

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For another important note while working with radicals:

If you have two square roots to add, you can't just add the radicands (the same goes for subtraction):
  _       __      ____
√9 + √16 ≠ √9+16
3 + 4 = 7, √9 + 16 = 5

However, you CAN multiply and divide:
  ___    __      ______
√144/√36 = √144/36
12 / 6 = 2, √4 = 2

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