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### Surd Trick

```Date: 02/14/2012 at 21:07:59
From: Praveen
Subject: Surd

Find the value of

SQRT[49 - 20 SQRT(6)]

In case the meaning of these nested radicals is not clear, please see this
image:

http://www.imagebam.com/image/c7505a174856941

I cannot reduce SQRT(49) to 7, since something else gets subtracted from
it first. So I tried letting ...

x = SQRT[49 - 20 SQRT(6)]

... and then squaring both the sides, but I still couldn't get the answer.

Next, I tried to move the "20" inside the SQRT(6), which also was no use.

Thanks a lot.

```

```
Date: 02/16/2012 at 19:50:36
From: Doctor Vogler
Subject: Re: Surd

Hi Praveen,

Thanks for writing to Dr. Math.

Here is a trick for taking square roots of quadratic surds.

You have something of the form

a - sqrt(b).

(In your case, a = 49, and b = 6*20^2.) We assume that b is not a square.
You want to find a square root of the form ...

n - sqrt(d)

... or possibly ...

sqrt(c) - sqrt(d).

So you are looking for rational solutions (in c and d, given a and b) that
solve this equation:

sqrt(c) - sqrt(d) = sqrt(a - sqrt(b))

Equivalently:

(sqrt(c) - sqrt(d))^2 = a - sqrt(b)

Well, the last equation simplifies to

c + d - 2*sqrt(c*d) = a - sqrt(b).

Since b is not a square, and a, b, c, and d are all rational numbers, the
only way there could possibly be a solution is if

c + d = a    and    2*sqrt(c*d) = sqrt(b).

In other words,

c + d = a
4*c*d = b.

Now here's the big leap: It turns out that if this happens, then

a^2 - b = (c - d)^2.

So if you check that a^2 - b is not a perfect square, then the square root
quartic. In this case, you can't simplify the square root.

But if this ...

a^2 - b = r^2

... is the square of some rational number, then you can declare
c = (a + r)/2
d = (a - r)/2

Now, you will get both ...

sqrt(c) - sqrt(d) = sqrt(a - sqrt(b))

... and ...

sqrt(c) + sqrt(d) = sqrt(a + sqrt(b)).

Of course, if c or d is itself a perfect square (as happens rather often),
then you can simplify further.

and show me what you have been able to do, and I will try to offer further
suggestions.

- Doctor Vogler, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Square Roots

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