Question 90237
Start with the expression

{{{(-9 - sqrt(108))/3}}}


First lets reduce {{{sqrt(108)}}}

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{{{sqrt(108)}}} Start with the given expression
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 108

Factors:

1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54,



Notice how 36 is the largest perfect square, so lets break 108 down into 36*3



{{{sqrt(36*3)}}} Factor 108 into 36*3
 
{{{sqrt(36)*sqrt(3)}}} Break up the square roots using the identity {{{sqrt(x*y)=sqrt(x)*sqrt(y)}}}
 
{{{6*sqrt(3)}}} Take the square root of the perfect square 36 to get 6 
 
So the expression


{{{sqrt(108)}}}


simplifies to


{{{6*sqrt(3)}}}

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{{{(-9 - 6*sqrt(3))/3}}} Simplify the square root (using the technique above)


{{{-9/3 - 6*sqrt(3)/3}}} Break up the fraction


{{{-3 - 6*sqrt(3)/3}}} Divide {{{-9/3}}} to get {{{-3}}}


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




So the expression

{{{(-9 - sqrt(108))/3}}}


simplifies to


{{{-3 - 2*sqrt(3)}}}



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Start with the expression

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


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

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{{{sqrt(20)}}} Start with the given expression
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,



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




So the expression

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


simplifies to


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