Question 176667

{{{sqrt(52)}}} 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 52



Factors:

1, 2, 4, 13, 26, 52



Notice how 4 is the largest perfect square, so lets factor 52 into 4*13



{{{sqrt(4*13)}}} Factor 52 into 4*13
 
 
{{{sqrt(4)*sqrt(13)}}} Break up the square roots using the identity {{{sqrt(x*y)=sqrt(x)*sqrt(y)}}}
 
 
{{{2*sqrt(13)}}} Take the square root of the perfect square 4 to get 2 
 
 
So the expression {{{sqrt(52)}}} simplifies to {{{2*sqrt(13)}}}


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Check:


Notice if we evaluate the square root of 52 with a calculator we get


{{{sqrt(52)=7.21110255092798}}}


and if we evaluate {{{2*sqrt(13)}}} we get


{{{2*sqrt(13)=7.21110255092798}}}


This shows that {{{sqrt(52)=2*sqrt(13)}}}. So this verifies our answer