SOLUTION: Simplify {{{sqrt(20)/5-1/sqrt(5)}}}

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Question 117122: Simplify

sqrt%2820%29%2F5-1%2Fsqrt%285%29
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
sqrt%2820%29%2F5-1%2Fsqrt%285%29 Start with the given expression



sqrt%2820%29%2F5-%281%2Fsqrt%285%29%29%28sqrt%285%29%2Fsqrt%285%29%29 Multiply the second fraction by sqrt%285%29%2Fsqrt%285%29. Doing this will make
the denominator a rational number


sqrt%2820%29%2F5-sqrt%285%29%2F%28sqrt%285%29sqrt%285%29%29 Combine the fractions




sqrt%2820%29%2F5-sqrt%285%29%2F5 Multiply sqrt%285%29sqrt%285%29 to get 5. Notice how sqrt%285%29sqrt%285%29=%28sqrt%285%29%29%5E2=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



%28sqrt%2820%29-sqrt%285%29%29%2F5 Combine the numerators



%28sqrt%284%2A5%29-sqrt%285%29%29%2F5 Factor 20 into 4*5



%28sqrt%284%29%2Asqrt%285%29-sqrt%285%29%29%2F5 Break up the roots using the identity sqrt%28x%2Ay%29=sqrt%28x%29sqrt%28y%29.


%282%2Asqrt%285%29-sqrt%285%29%29%2F5 Take the square root of the perfect square 4 to get 2

Notice how we have the common term sqrt%285%29

%28%282-1%29%2Asqrt%285%29%29%2F5 Combine like terms


sqrt%285%29%2F5 Combine like terms



So sqrt%2820%29%2F5-1%2Fsqrt%285%29 simplifies to sqrt%285%29%2F5. In other words, sqrt%2820%29%2F5-1%2Fsqrt%285%29=sqrt%285%29%2F5.


So looking at the answer you wrote, you are correct since %281%2F5%29sqrt%285%29 is equivalent to sqrt%285%29%2F5

Check:


If we evaluate sqrt%2820%29%2F5-1%2Fsqrt%285%29 with a calculator, we get

sqrt%2820%29%2F5-1%2Fsqrt%285%29=0.44721359549996


and if we evaluate sqrt%285%29%2F5 with a calculator, we get

sqrt%285%29%2F5=0.44721359549996



Notice how these evaluate to the same value. So this shows that sqrt%2820%29%2F5-1%2Fsqrt%285%29=sqrt%285%29%2F5 and this verifies our
answer.