Question 103977


Start with the expression

{{{(50 + sqrt(125))/5}}}


First lets reduce {{{sqrt(125)}}}

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{{{sqrt(125)}}} 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 125

Factors:

1, 5, 25, 125



Notice how 25 is the largest perfect square, so lets break 125 down into 25*5



{{{sqrt(25*5)}}} Factor 125 into 25*5
 
{{{sqrt(25)*sqrt(5)}}} Break up the square roots using the identity {{{sqrt(x*y)=sqrt(x)*sqrt(y)}}}
 
{{{5*sqrt(5)}}} Take the square root of the perfect square 25 to get 5 
 
So the expression


{{{sqrt(125)}}}


simplifies to


{{{5*sqrt(5)}}}

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{{{(50 + 5*sqrt(5))/5}}} Simplify the square root (using the technique above)


{{{50/5 + 5*sqrt(5)/5}}} Break up the fraction


{{{10 + 5*sqrt(5)/5}}} Divide {{{50/5}}} to get {{{10}}}


{{{10 + 1*sqrt(5)}}} Divide {{{5/5}}} to get {{{1}}}




So the expression

{{{(50 + sqrt(125))/5}}}


simplifies to


{{{10 + sqrt(5)}}}