```Question 350761

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

If you need more help, email me at <a href="mailto:jim_thompson5910@hotmail.com?Subject=Algebra%20Help">jim_thompson5910@hotmail.com</a>

Also, feel free to check out my <a href="http://www.freewebs.com/jimthompson5910/home.html">tutoring website</a>

Jim```