Question

. Solve: x^2/ 3 − 2x ^1/ 3 = 15

Answer 1

Answer:

x*2/3- 2x*1/3

x-2x*2/4*1/3

-x*6+4/12

-x*10/12 =15

15+x=10/12

15+x=5/6

3+x=1/6

1 +x = 1/2

x = 1/2-1

x = -1/2

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Answer 2

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

TO SOLVE

$\displaystyle \: {x}^{ \frac{2}{3} } - 2 {x}^{ \frac{1}{3} } = 15$

EVALUATION

$\sf{\displaystyle \: {x}^{ \frac{2}{3} } - 2 {x}^{ \frac{1}{3} } = 15 }\: \: \: ...(1)$

$let \: y \: \displaystyle = \: {x}^{ \frac{1}{3} }$

Then Equation (1) becomes

${y}^{2} - 2y = 15$

$\implies \: {y}^{2} - 5y + 3y - 15 = 0$

$\implies \: y(y - 5) + 3(y - 5 )= 0$

$\implies \: (y - 5) (y + 3 )= 0$

So

$either \: (y - 5) = 0 \: \: or \: \: (y + 3 )= 0$

$y - 5 = 0 \: \: implies \: \: y = 5$

$\implies \: \: \displaystyle \: {x}^{ \frac{1}{3} } = 5$

$\implies \: \: \displaystyle \: x = {5}^{3}$

$\implies \: \: \displaystyle \: x = 125$

Again

$y + 3= 0 \: \: implies \: \: y = - 3$

$\implies \: \: \displaystyle \: {x}^{ \frac{1}{3} } = - 3$

$\implies \: \: \displaystyle \: x = {( - 3)}^{3}$

$\implies \: \: \displaystyle \: x = - 27$

RESULT

SO the required solution is

$\boxed{ \:x = - 27 \: , \: 125 \: }$