Question 86162
{{{(18+sqrt(567))/9}}}


First lets reduce {{{sqrt(567)}}}

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{{{sqrt(567)}}} 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 567

Factors:

1, 3, 7, 9, 21, 27, 63, 81, 189,



Notice how 81 is the largest perfect square, so lets break 567 down into 81*7



{{{sqrt(81*7)}}} Factor 567 into 81*7
 
{{{sqrt(81)*sqrt(7)}}} Break up the square roots using the identity {{{sqrt(x*y)=sqrt(x)*sqrt(y)}}}
 
{{{9*sqrt(7)}}} Take the square root of the perfect square 81 to get 9 
 
So the expression


{{{sqrt(567)}}}


simplifies to


{{{9*sqrt(7)}}}

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{{{(18+9*sqrt(7))/9}}}  Simplify the square root (using the technique above)


{{{18/9+(9*sqrt(7))/9}}} Break up the fraction


{{{2+(9*sqrt(7))/9}}} Divide {{{18/9}}} to get 2


{{{2+1*sqrt(7)}}} Divide {{{9/9}}} to get 1


So the expression 


{{{(18+sqrt(567))/9}}}


Simplifies to 


{{{2+sqrt(7)}}}