Given that \[ x^3+\frac{1}{x^3}=4, \] find the value of \[ P=x^6+\frac{4}{x^3}. \]

Solve the Algebraic Problem

Given that

\[ x^3+\frac{1}{x^3}=4, \]

find the value of

\[ P=x^6+\frac{4}{x^3}. \]

---

Solution

Step 1. Introduce a New Variable

Let

\[ a=x^3. \]

Then the given equation becomes

\[ a+\frac{1}{a}=4. \]

Step 2. Form a Quadratic Equation

Multiply both sides by \(a\):

\[ a^2+1=4a. \]

Hence,

\[ a^2-4a+1=0. \]

Therefore,

\[ a^2=4a-1. \]

Step 3. Evaluate the Required Expression

Since

\[ P=x^6+\frac{4}{x^3}=a^2+\frac{4}{a}, \]

and

\[ \frac{1}{a}=4-a, \]

we obtain

\[ \begin{aligned} P &=a^2+4(4-a)\\ &=a^2-4a+16. \end{aligned} \]

Substitute \(a^2=4a-1\):

\[ \begin{aligned} P &=(4a-1)-4a+16\\ &=15. \end{aligned} \]

---

Final Answer

\[ \boxed{15} \]