Question 140637

Start with the expression

{{{(-64 + sqrt(4288))/-32}}}


First lets reduce {{{sqrt(4288)}}}

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{{{sqrt(4288)}}} 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 4288

Factors:

1, 2, 4, 8, 16, 32, 64, 67, 134, 268, 536, 1072, 2144, 4288



Notice how 64 is the largest perfect square, so lets break 4288 down into 64*67



{{{sqrt(64*67)}}} Factor 4288 into 64*67
 
{{{sqrt(64)*sqrt(67)}}} Break up the square roots using the identity {{{sqrt(x*y)=sqrt(x)*sqrt(y)}}}
 
{{{8*sqrt(67)}}} Take the square root of the perfect square 64 to get 8 
 
So the expression


{{{sqrt(4288)}}}


simplifies to


{{{8*sqrt(67)}}}

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{{{(-64 + 8*sqrt(67))/-32}}} Simplify the square root (using the technique above)


{{{-64/-32 + 8*sqrt(67)/-32}}} Break up the fraction


{{{2 + 8*sqrt(67)/-32}}} Divide {{{-64/-32}}} to get {{{2}}}


{{{2 -(1/4)*sqrt(67)}}} Divide {{{8/-32}}} to get {{{-1/4}}}




So the expression

{{{(-64 + sqrt(4288))/-32}}}


simplifies to


{{{2 -(1/4)*sqrt(67)}}}