Question

if 2x+1/3x=4 then prove that 27x^3+1/8x^3=189

Answer 1

I think your question is wrong

Answer 2

Answer:  The proof is given below.

Step-by-step explanation:  We are given that

$2x+\dfrac{1}{3x}=4~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)$

We are to prove the following :

$27x^3+\dfrac{1}{8x^3}=189.$

We will be using the following factorization formula :

$(a+b)^3=a^3+b^3+3ab(a+b).$

We have from equation (i) that

$2x+\dfrac{1}{3x}=4\\\\\\\Rightarrow x+\dfrac{1}{6x}=2~~~~~~~~~~~~~[\textup{divising both sides by 2}]\\\\\\\Rightarrow 3x+\dfrac{1}{2x}=6~~~~~~~~~~~[\textup{multiplying both sides by 3}]\\\\\\\Rightarrow \left(3x+\dfrac{1}{2x}\right)^3=6^3~~~~~~~~~~~~[\textup{Cubing borh sides}]\\\\\\\Rightarrow (3x)^3+\left(\dfrac{1}{2x}\right)^2+3\times 3x\times\dfrac{1}{2x}\left(3x+\dfrac{1}{2x}\right)=216\\\\\\\Rightarrow 27x^3+\dfrac{1}{8x^3}+\dfrac{9}{2}\times6=216\\\\\\\Rightarrow 27x^3+\dfrac{1}{8x^3}+27=216\\\\\\\Rightarrow 27x^3+\dfrac{1}{8x^3}=216-27\\\\\\\Rightarrow 27x^3+\dfrac{1}{8x^3}=189.$

Hence proved.