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If a and b are real numbers, \(a^2b^3=\frac{32}{27}\), and \(\frac{a}{b^3}=\frac{27}{4}\), what is a+b?

 Mar 3, 2019
 #1
avatar+99582 
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a^2 b^3 = 32/27     (1)      

a/b^3 =  27/ 4    ⇒    b^3 =  (4/27) a     (2)

 

Sub (2) into (1)  and we have that

 

a^2 (4/27) a =   32/27     simplify

 

a^3 (4/27) = 32/27

 

a^3 = ( 32/27) (27/4)

 

a^3 = 32/4 

 

a^3 = 8

 

a = 2     and   

 

b^3 = (4/27) (2)    =   8 /27

 

So....taking the cube root of both sides

 

b = 2/3

 

So

 

a + b  =  2 + 2/3  =  8 / 3

 

 

cool cool cool

 Mar 3, 2019

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