Question 570944


{{{sqrt(686)}}} 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. When you take the square root of this perfect square, you will get a rational number.



So let's list the factors of 686



Factors:

1, 2, 7, 14, 49, 98, 343, 686



Notice how 49 is the largest perfect square, so lets factor 686 into 49*14



{{{sqrt(49*14)}}} Factor 686 into 49*14
 
{{{sqrt(49)*sqrt(14)}}} Break up the square roots using the identity {{{sqrt(x*y)=sqrt(x)*sqrt(y)}}}
 
{{{7*sqrt(14)}}} Take the square root of the perfect square 49 to get 7 
 
So the expression {{{sqrt(686)}}} simplifies to {{{7*sqrt(14)}}}


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Check:

Notice if we evaluate the square root of 686 with a calculator we get


{{{sqrt(686)=26.1916017074176}}}


and if we evaluate {{{7*sqrt(14)}}} we get


{{{7*sqrt(14)=26.1916017074176}}}


This shows that {{{sqrt(686)=7*sqrt(14)}}}. So this verifies our answer