```Question 118070

{{{(6 - sqrt(20))/2}}}

First lets reduce {{{sqrt(20)}}}

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The goal of simplifying expressions with square roots is to factor the radicand into a product of two numbers. One of these two numbers must be a perfect square. This way the perfect square will become a rational number.

So let's list the factors of 20

Factors:

1, 2, 4, 5, 10, 20

Notice how 4 is the largest perfect square, so lets break 20 down into 4*5

{{{sqrt(4*5)}}} Factor 20 into 4*5

{{{sqrt(4)*sqrt(5)}}} Break up the square roots using the identity {{{sqrt(x*y)=sqrt(x)*sqrt(y)}}}

{{{2*sqrt(5)}}} Take the square root of the perfect square 4 to get 2

So the expression

{{{sqrt(20)}}}

simplifies to

{{{2*sqrt(5)}}}

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{{{(6 - 2*sqrt(5))/2}}} Simplify the square root (using the technique above)

{{{6/2 - 2*sqrt(5)/2}}} Break up the fraction

{{{3 - 2*sqrt(5)/2}}} Divide {{{6/2}}} to get {{{3}}}

{{{3 - 1*sqrt(5)}}} Divide {{{2/2}}} to get {{{1}}}

So the expression

{{{(6 - sqrt(20))/2}}}

simplifies to

{{{3 - sqrt(5)}}}
```