Question 119144


{{{sqrt(75)}}} 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 75



Factors:

1, 3, 5, 15, 25, 75



Notice how 25 is the largest perfect square, so lets factor 75 into 25*3



{{{sqrt(25*3)}}} Factor 75 into 25*3
 
{{{sqrt(25)*sqrt(3)}}} Break up the square roots using the identity {{{sqrt(x*y)=sqrt(x)*sqrt(y)}}}
 
{{{5*sqrt(3)}}} Take the square root of the perfect square 25 to get 5 
 
So the expression {{{sqrt(75)}}} simplifies to {{{5*sqrt(3)}}}


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Check:

Notice if we evaluate the square root of 75 with a calculator we get


{{{sqrt(75)=8.66025403784439}}}


and if we evaluate {{{5*sqrt(3)}}} we get


{{{5*sqrt(3)=8.66025403784439}}}


This shows that {{{sqrt(75)=5*sqrt(3)}}}. So this verifies our answer