Question

If \[{{x}^{\dfrac{2}{3}}}-7{{x}^{\dfrac{1}{3}}}+10=0\], then the value of x is
1. \[\dfrac{13}{32}\]
2. \[\dfrac{1}{4}\]
3. \[\dfrac{1}{32}\]
4. \[\dfrac{3}{16}\]

Answer 1

Hint: Now to solve this question you first start by taking a different variable in the place of \[{{x}^{\dfrac{1}{3}}}\] on doing that we can see that the equation we get after it will be similar to the general equal of quadratic equation which is \[a{{y}^{2}}+by+c=0\]. Thus we can now through this get the value of a,b and c by solving the quadratic equation. After that putting the values in the assumed variables we can get the values of for x.

Complete step-by-step answer:
First we have gotten the equation that \[{{x}^{\dfrac{2}{3}}}-7{{x}^{\dfrac{1}{3}}}+10=0\]. Now to simplify this we can take \[{{x}^{\dfrac{1}{3}}}\] as y which will help us in solving this question further. Therefore the equation now will be
\[{{y}^{2}}-7y+10=0\]
Now this is a general equation of any quadratic equation which usually is in the form of \[a{{y}^{2}}+by+c=0\] . Now on comparing both of the equations with each other we can get the values of a,b and c. The values of a, b and c are
\[a=1,b=-7,c=10\]
Now to solve this we can use the standard formula of quadratic equations which we use to find the value of the variable. The formula is
\[y=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]
Substituting the values of a , b and c we get
\[y=\dfrac{7\pm \sqrt{{{7}^{2}}-4\times 1\times 10}}{2}\]
Solving the root
\[y=\dfrac{7\pm \sqrt{9}}{2}\]
Opening the root we get two values of y which are
\[y=\dfrac{7+3}{2}\] ; \[y=\dfrac{7-3}{2}\]
Simplifying
\[y=5,2\]
Now we also know that
\[{{x}^{{}^{1}/{}_{3}}}=y\]
Therefore
\[{{x}^{{}^{1}/{}_{3}}}=5\] ; \[{{x}^{{}^{1}/{}_{3}}}=2\]
Now cubing both sides we get the two values of x which are
\[x=125,8\]
Hence the values of x for the given equation are \[x=125,8\]
So, the correct answer is “\[x=125,8\]”.

Note: We can also just simplify the equation and find the roots by using the sum and product method but the easiest method is to just put the values of a,b and c in the quadratic equation formula