Question 101751
{{{root(3,(2z+3)^2)+root(3,(2z+3))=6}}} Start with the given equation


Let {{{y=2z+3}}}



{{{root(3,y^2)+root(3,y)=6}}} So now we get this equation in terms of y



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Here's an important side note: Take {{{root(3,y^2)+root(3,y)=6}}} and subtract {{{root(3,y)}}} from both sides to get {{{root(3,y^2)=6-root(3,y)}}}. This will be useful later

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Now let's move back to the main problem:


{{{root(3,y*y)+root(3,y)=6}}} Factor {{{y^2}}}



{{{root(3,y)*root(3,y)+root(3,y)=6}}} Break down the root



{{{root(3,y)(root(3,y)+1)=6}}} Factor out {{{root(3,y)}}}



{{{(root(3,y)+1)=6/root(3,y)}}} Now divide both sides by {{{root(3,y)}}}



{{{(root(3,y)+1)^3=(6/root(3,y))^3}}} Cube both sides



{{{y+3root(3,y)+3root(3,y^2)+1=216/y}}} Expand the left side



Remember, {{{root(3,y^2)=6-root(3,y)}}}, so replace {{{root(3,y^2)}}} with {{{6-root(3,y)}}}



{{{y+3root(3,y)+3*highlight((6-root(3,y)))+1=216/y}}} 



{{{y+3root(3,y)+18-3root(3,y)+1=216/y}}} Distribute



{{{y+18+cross(3root(3,y)-3root(3,y))+1=216/y}}} Notice the terms {{{3root(3,y)}}} and {{{-3root(3,y)}}} cancel to zero



{{{y+18+1=216/y}}} 



{{{y+19=216/y}}} Combine like terms




{{{y^2+19y^2=216}}} Multiply both sides by the LCD y to clear any fractions




{{{y^2+19y^2-216=0}}} Get all terms to one side




Now simply use any technique to solve for y



When you solve for y (I simply used a calculator), you get 


{{{y=8}}} or {{{y=-27}}}



Now plug in the first answer {{{y=8}}} into  {{{y=2z+3}}}



{{{8=2z+3}}}



Now solve for z


{{{z=5/2}}} which is {{{z=2.5}}}



Now plug in the second answer {{{y=-27}}} into  {{{y=2z+3}}}



{{{-27=2z+3}}}



Now solve for z


{{{z=-15}}} 





Check:



Let's check the answer {{{z=5/2}}}

{{{root(3,(2z+3)^2)+root(3,(2z+3))=6}}} Start with the given equation




{{{root(3,(2(5/2)+3)^2)+root(3,(2(5/2)+3))=6}}} Plug in {{{z=5/2}}}




{{{root(3,(5+3)^2)+root(3,(5+3))=6}}} Multiply 2 and {{{5/2}}} to get 5




{{{root(3,8^2)+root(3,8)=6}}} Add 5 and 3 to get 8



{{{root(3,64)+root(3,8)=6}}} Square 8 to get 64




{{{4+2=6}}} Take the cube root of 64 to get 4 and take the cube root of 8 to get 2



{{{6=6}}} Since the two sides of the equation are equal, this verifies our answer.






Now let's check the answer {{{z=-15}}}

{{{root(3,(2z+3)^2)+root(3,(2z+3))=6}}} Start with the given equation




{{{root(3,(2(-15)+3)^2)+root(3,(2(-15)+3))=6}}} Plug in {{{z=-15}}}




{{{root(3,(-30+3)^2)+root(3,(-30+3))=6}}} Multiply 2 and {{{-15}}} to get -30




{{{root(3,(-27)^2)+root(3,-27)=6}}} Add -30 and 3 to get -27



{{{root(3,729)+root(3,-27)=6}}} Square -27 to get 729




{{{9-3=6}}} Take the cube root of 729 to get 9 and take the cube root of -27 to get -3



{{{6=6}}} Since the two sides of the equation are equal, this verifies our answer.