Question

If we have an expression \[p = {2^{\dfrac{2}{3}}} + {2^{\dfrac{1}{3}}}\], then?
A). \[{p^3} - 6p + 6 = 0\]
B). \[{p^3} - 3p - 6 = 0\]
C). \[{p^3} - 6p - 6 = 0\]
D). \[{p^3} - 3p + 6 = 0\]

Answer 1

Hint: In the given question, we have been given an expression in terms of a variable equal to constants raised to some power. We have to calculate the value of the variable in terms of quadratic equations. We are going to solve it by taking common, simplifying the values, solving the commons and finding the value.

Complete step by step solution:
The given equation is \[p = {2^{\dfrac{2}{3}}} + {2^{\dfrac{1}{3}}}\].
Taking cube on both sides,
\[{p^3} = {\left( {{2^{\dfrac{2}{3}}}} \right)^3} + {\left( {{2^{\dfrac{1}{3}}}} \right)^3} + 3 \times {2^{\dfrac{2}{3}}} \times {2^{\dfrac{1}{3}}}\left( {{2^{\dfrac{1}{3}}} + {2^{\dfrac{2}{3}}}} \right)\]
Simplifying the brackets,
\[{p^3} = 4 + 2 + 3 \times 2 \times p\]
Hence, \[{p^3} - 6p - 6 = 0\]
Thus, the correct option is C.

Note: In this question, we were given an expression in terms of a variable equal to constants raised to some power. We had to calculate the value of the variable in terms of the quadratic equation. We solved it by taking common, simplifying the values, solving the common and finding the value. So, it is very important that we know how to solve the equations, how to deal with the radicals, how to take commons and how to simplify the values.