Question 119793

Start with the expression

{{{(12 + sqrt(108))/6}}}


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, 108



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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{{{(12 + 6*sqrt(3))/6}}} Simplify the square root (using the technique above)


{{{12/6 + 6*sqrt(3)/6}}} Break up the fraction


{{{2 + 6*sqrt(3)/6}}} Divide {{{12/6}}} to get {{{2}}}


{{{2 + 1*sqrt(3)}}} Divide {{{6/6}}} to get {{{1}}}




So the expression

{{{(12 + sqrt(108))/6}}}


simplifies to


{{{2 + sqrt(3)}}}