Question 117122
{{{sqrt(20)/5-1/sqrt(5)}}} Start with the given expression




{{{sqrt(20)/5-(1/sqrt(5))(sqrt(5)/sqrt(5))}}} Multiply the second fraction by {{{sqrt(5)/sqrt(5)}}}. Doing this will make 

the denominator a rational number



{{{sqrt(20)/5-sqrt(5)/(sqrt(5)sqrt(5))}}} Combine the fractions





{{{sqrt(20)/5-sqrt(5)/5}}} Multiply {{{sqrt(5)sqrt(5)}}} to get {{{5}}}. Notice how {{{sqrt(5)sqrt(5)=(sqrt(5))^2=5}}} shows 

how the square undoes the square root.



Now that we have a rational denominator, and also a common denominator, we can combine the fractions




{{{(sqrt(20)-sqrt(5))/5}}} Combine the numerators




{{{(sqrt(4*5)-sqrt(5))/5}}} Factor 20 into 4*5




{{{(sqrt(4)*sqrt(5)-sqrt(5))/5}}} Break up the roots using the identity {{{sqrt(x*y)=sqrt(x)sqrt(y)}}}.



{{{(2*sqrt(5)-sqrt(5))/5}}} Take the square root of the perfect square 4 to get 2


Notice how we have the common term {{{sqrt(5)}}}


{{{((2-1)*sqrt(5))/5}}} Combine like terms



{{{sqrt(5)/5}}} Combine like terms




So {{{sqrt(20)/5-1/sqrt(5)}}} simplifies to {{{sqrt(5)/5}}}. In other words, {{{sqrt(20)/5-1/sqrt(5)=sqrt(5)/5}}}. 



So looking at the answer you wrote, you are correct since {{{(1/5)sqrt(5)}}} is equivalent to {{{sqrt(5)/5}}}


Check:



If we evaluate {{{sqrt(20)/5-1/sqrt(5)}}} with a calculator, we get 


{{{sqrt(20)/5-1/sqrt(5)=0.44721359549996}}}



and if we evaluate {{{sqrt(5)/5}}} with a calculator, we get 


{{{sqrt(5)/5=0.44721359549996}}}




Notice how these evaluate to the same value. So this shows that {{{sqrt(20)/5-1/sqrt(5)=sqrt(5)/5}}} and this verifies our 

answer.