Question 118001
#1


{{{5*sqrt(72)}}} Start with the given expression



Let's simplify {{{sqrt(72)}}}


{{{sqrt(72)}}} 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 72

Factors:

1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72



Notice how 36 is the largest perfect square, so lets factor 72 into 36*2



{{{sqrt(36*2)}}} Factor 72 into 36*2
 
{{{sqrt(36)*sqrt(2)}}} Break up the square roots using the identity {{{sqrt(x*y)=sqrt(x)*sqrt(y)}}}
 
{{{6*sqrt(2)}}} Take the square root of the perfect square 36 to get 6 
 
So the expression {{{sqrt(72)}}} simplifies to {{{6*sqrt(2)}}}




Now reintroduce the 5 to get {{{5*6*sqrt(2)}}}. Now multiply to get


{{{30*sqrt(2)}}}



So {{{5*sqrt(72)}}} simplifies to {{{30*sqrt(2)}}}


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Check:

Notice if we evaluate {{{5*sqrt(72)}}} with a calculator we get


{{{5*sqrt(72)=42.4264068711929}}}


and if we evaluate {{{30*sqrt(2)}}} we get


{{{30*sqrt(2)=42.4264068711929}}}


This shows that {{{5*sqrt(72)=30*sqrt(2)}}}. So this verifies our answer 



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#2

{{{sqrt(120)+sqrt(75)}}} Start with the given expression



{{{sqrt(15*8)+sqrt(15*5)}}} Factor 120 into 15*8 and factor 75 into 15*5



{{{sqrt(15)*sqrt(8)+sqrt(15)*sqrt(5)}}} Break up the square roots




{{{sqrt(15)(sqrt(8)+sqrt(5))}}} Factor out the common term of {{{sqrt(15)}}}



So {{{sqrt(120)+sqrt(75)}}} simplifies to {{{sqrt(15)(sqrt(8)+sqrt(5))}}}