Question 86391
a)


{{{sqrt(12)+sqrt(108)}}}


First lets simplify {{{sqrt(12)}}}:



{{{sqrt(12)}}} 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. This way the perfect square will become a rational number.

So let's list the factors of 12

Factors:

1, 2, 3, 4, 6,



Notice how 4 is the largest perfect square, so lets break 12 down into 4*3



{{{sqrt(4*3)}}} Factor 12 into 4*3
 
{{{sqrt(4)*sqrt(3)}}} Break up the square roots using the identity {{{sqrt(x*y)=sqrt(x)*sqrt(y)}}}
 
{{{2*sqrt(3)}}} Take the square root of the perfect square 4 to get 2 
 
So the expression


{{{sqrt(12)}}}


simplifies to


{{{2*sqrt(3)}}}

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Now lets simplify {{{sqrt(108)}}}:


 
{{{sqrt(108)}}} 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. This way the perfect square will become a rational number.

So let's list the factors of 108

Factors:

1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54,



Notice how 36 is the largest perfect square, so lets break 108 down into 36*3



{{{sqrt(36*3)}}} Factor 108 into 36*3
 
{{{sqrt(36)*sqrt(3)}}} Break up the square roots using the identity {{{sqrt(x*y)=sqrt(x)*sqrt(y)}}}
 
{{{6*sqrt(3)}}} Take the square root of the perfect square 36 to get 6 
 
So the expression


{{{sqrt(108)}}}


simplifies to


{{{6*sqrt(3)}}}



So the expression


{{{sqrt(12)+sqrt(108)}}}


simplifies to 


{{{2*sqrt(3)+6*sqrt(3)}}}



Notice we have a common term of {{{sqrt(3)}}}. If we let {{{y=sqrt(3)}}} we get


{{{2y+6y}}} 


Now combine like terms


{{{8y}}}}


Replace y with {{{sqrt(3)}}}


{{{8*sqrt(3)}}}



Check:


Evaluate the given expression with a calculator:


{{{sqrt(12)+sqrt(108)=13.856406460551}}}


Evaluate the simplified expression with a calculator:


{{{8*sqrt(3)=13.856406460551}}}


Since they are equal (to a certain decimal place), this verifies our answer




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b)


{{{root(3,16)*root(3,3)}}}


Since the 2 radicands are under the same root value, we can combine them using the identity {{{root(n,x)*root(n,y)=root(n,x*y)}}}


{{{root(3,16*3)}}} Combine the cube roots


{{{root(3,48)}}} Multiply


{{{root(3,8*6)}}} Factor 48 into 8*6. I chose to factor out an 8 since 8 is a perfect cube


{{{root(3,8)*root(3,6)}}} Break up the roots using {{{root(n,x*y)=root(n,x)*root(n,y)}}}


{{{2*root(3,6)}}} Take the cube root of 8


Check:


Evaluate the given expression with a calculator:


{{{root(3,16)*root(3,3)=3.63424118566428}}}


Evaluate the simplified expression with a calculator:


{{{2*root(3,6)=3.63424118566428}}} 


Since they are equal (to a certain decimal place), this verifies our answer