Question 118059
#1


{{{sqrt(50)}}} 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 50

Factors:

1, 2, 5, 10, 25, 50



Notice how 25 is the largest perfect square, so lets factor 50 into 25*2



{{{sqrt(25*2)}}} Factor 50 into 25*2
 
{{{sqrt(25)*sqrt(2)}}} Break up the square roots using the identity {{{sqrt(x*y)=sqrt(x)*sqrt(y)}}}
 
{{{5*sqrt(2)}}} Take the square root of the perfect square 25 to get 5 
 
So the expression {{{sqrt(50)}}} simplifies to {{{5*sqrt(2)}}}


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Check:

Notice if we evaluate the square root of 50 with a calculator we get


{{{sqrt(50)=7.07106781186548}}}


and if we evaluate {{{5*sqrt(2)}}} we get


{{{5*sqrt(2)=7.07106781186548}}}


This shows that {{{sqrt(50)=5*sqrt(2)}}}. So this verifies our answer 



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#2



{{{sqrt(72*x^3)}}} Start with the given expression

{{{sqrt(36*2*x^3)}}} Factor {{{72}}} into {{{36*2}}}
 
{{{sqrt(36*2*x*x^2)}}} Factor {{{x^3}}} into {{{x*x^2}}}
 
{{{sqrt(36)*sqrt(2)*sqrt(x)*sqrt(x^2)}}} Break up the square root using the identity {{{sqrt(x*y)=sqrt(x)*sqrt(y)}}}
 
{{{6*sqrt(2)*sqrt(x)*sqrt(x^2)}}} Take the square root of the perfect square {{{36}}} to get 6 
 
{{{6*sqrt(2)*sqrt(x)*x}}} Take the square root of the perfect squares {{{x^2}}} to get {{{x}}} 
 

{{{6*x*sqrt(x)*sqrt(2)}}} Rearrange the terms 

{{{6*x*sqrt(2x)}}} Group the square root terms 



So {{{sqrt(72*x^3)}}} simplifies to {{{6*x*sqrt(2x)}}}