Question

Solve :- $x^{\frac{2}{3}}-5x^{\frac{1}{3}}-14=0$

Short answer

Solution for equation will be -

• $x=343$
  
• $x=(-8)$
  

Step-by-step solution

Step 1.  Given Data 

Quadratic Equation - $x^{\frac{2}{3}}-5x^{\frac{1}{3}}-14=0$

Step 2.  To Find  

We have to solve it and find the value of x.

Step 3.  Explanation  

Let the value of $x^{\frac{1}{3}}=u$

Then the equation is -

$x^{\frac{2}{3}}-5x^{\frac{1}{3}}-14=0$

${\left(x^{\frac{1}{3}}\right)}^2-5x^{\frac{1}{3}}-14=0$

Put the value of $x^{\frac{1}{3}}=u$

$u^2-5u-14=0$

$(u-7)(u+2)=0$

By using zero product property -

$u-7=0$, $u+2=0$

$u=7$, role="math" localid="1645025440134" $u=(-2)$

Step 4.  Calculation 

Replace u with $x^{\frac{1}{3}}$

$x^{\frac{1}{3}}=7$, $x^{\frac{1}{3}}=(-2)$

Solve for x by cubing both sides

${\left(x^{\frac{1}{3}}\right)}^3=7^3$, ${\left(x^{\frac{1}{3}}\right)}^3={(-2)}^3$

$x=343$, $x=(-8)$