Question 120455

Start with the expression

{{{(-4 + sqrt(40))/-2}}}


First lets reduce {{{sqrt(40)}}}

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{{{sqrt(40)}}} 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 40

Factors:

1, 2, 4, 5, 8, 10, 20, 40



Notice how 4 is the largest perfect square, so lets break 40 down into 4*10



{{{sqrt(4*10)}}} Factor 40 into 4*10
 
{{{sqrt(4)*sqrt(10)}}} Break up the square roots using the identity {{{sqrt(x*y)=sqrt(x)*sqrt(y)}}}
 
{{{2*sqrt(10)}}} Take the square root of the perfect square 4 to get 2 
 
So the expression


{{{sqrt(40)}}}


simplifies to


{{{2*sqrt(10)}}}

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{{{(-4 + 2*sqrt(10))/-2}}} Simplify the square root (using the technique above)


{{{-4/-2 + 2*sqrt(10)/-2}}} Break up the fraction


{{{2 + 2*sqrt(10)/-2}}} Divide {{{-4/-2}}} to get {{{2}}}


{{{2 + -1*sqrt(10)}}} Divide {{{2/-2}}} to get {{{-1}}}



{{{2-sqrt(10)}}} Multiply




So the expression

{{{(-4 + sqrt(40))/-2}}}


simplifies to


{{{2-sqrt(10)}}}