Question 115070
{{{sqrt(6000)}}} 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 6000

Factors:

1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 125, 150, 200, 240, 250, 300, 375, 400, 500, 600, 750, 1000, 1200, 1500, 2000, 3000, 6000



Notice how 400 is the largest perfect square, so lets factor 6000 into 400*15



{{{sqrt(400*15)}}} Factor 6000 into 400*15
 
{{{sqrt(400)*sqrt(15)}}} Break up the square roots using the identity {{{sqrt(x*y)=sqrt(x)*sqrt(y)}}}
 
{{{20*sqrt(15)}}} Take the square root of the perfect square 400 to get 20 
 
So the expression {{{sqrt(6000)}}} simplifies to {{{20*sqrt(15)}}}


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Check:

Notice if we evaluate the square root of 6000 with a calculator we get


{{{sqrt(6000)=77.4596669241483}}}


and if we evaluate {{{20*sqrt(15)}}} we get


{{{20*sqrt(15)=77.4596669241483}}}


This shows that {{{sqrt(6000)=20*sqrt(15)}}}. So this verifies our answer