Certainly! Let's work through the problem step by step.
Given:
[tex]\[ x = 2^{1/3} + 2^{2/3} \][/tex]
We need to prove that:
[tex]\[ x^3 - 6x = 6 \][/tex]
### Step 1: Calculate [tex]\( x^3 \)[/tex]
First, let's compute [tex]\( x^3 \)[/tex]. We start with:
[tex]\[ x = 2^{1/3} + 2^{2/3} \][/tex]
Calculating [tex]\( x^3 \)[/tex] involves expanding the binomial:
[tex]\[ x^3 = \left( 2^{1/3} + 2^{2/3} \right)^3 \][/tex]
Using the binomial theorem, we get:
[tex]\[ x^3 = (2^{1/3})^3 + 3(2^{1/3})^2(2^{2/3}) + 3(2^{1/3})(2^{2/3})^2 + (2^{2/3})^3 \][/tex]
Since [tex]\((2^{1/3})^3 = 2\)[/tex] and [tex]\((2^{2/3})^3 = 4\)[/tex], we simplify the other terms:
[tex]\[ x^3 = 2 + 3 \cdot 2^{2/3} \cdot 2 = 2 + 6 \cdot 2^{2/3} + 3 \cdot 2 \cdot 2^{1/3} = 2 + 6 \cdot 2^{2/3} + 6 \cdot 2^{1/3} + 4 \][/tex]
Combining like terms, we get:
[tex]\[ x^3 = 2 + 4 + 6 \cdot 2^{1/3} + 6 \cdot 2^{2/3} \][/tex]
[tex]\[ x^3 = 6 + 6 \left( 2^{1/3} + 2^{2/3} \right) \][/tex]
Since [tex]\( x = 2^{1/3} + 2^{2/3} \)[/tex], we have:
[tex]\[ x^3 = 6 + 6x \][/tex]
### Step 2: Rearrange the equation
We need to prove that [tex]\( x^3 - 6x = 6 \)[/tex]. Rearrange the equation we found:
[tex]\[ x^3 = 6 + 6x \][/tex]
Subtract [tex]\( 6x \)[/tex] from both sides:
[tex]\[ x^3 - 6x = 6 \][/tex]
### Conclusion
We have successfully shown through the computation of [tex]\( x^3 \)[/tex] and rearranging the result, that:
[tex]\[ x^3 - 6x = 6 \][/tex]
Hence, the proof is complete:
[tex]\[ x^3 - 6x = 6 \][/tex]
