Certainly! Let's solve this problem step by step.
### Step 1: Understanding the Given Equation
We are given the equation:
[tex]\[ x^2 = 3^{\frac{2}{3}} + 3^{\frac{-2}{3}} - 2 \][/tex]
First, let's calculate the right-hand side:
1. [tex]\( 3^{\frac{2}{3}} \)[/tex] denotes the cube root of [tex]\(3\)[/tex] squared:
[tex]\[ 3^{\frac{2}{3}} = \left( 3^{\frac{1}{3}} \right)^2 \][/tex]
2. [tex]\( 3^{-\frac{2}{3}} \)[/tex] is the reciprocal of [tex]\(3^{\frac{2}{3}}\)[/tex]:
[tex]\[ 3^{-\frac{2}{3}} = \frac{1}{3^{\frac{2}{3}}} \][/tex]
Now let's compute the expression:
[tex]\[ 3^{\frac{2}{3}} + 3^{-\frac{2}{3}} - 2 \][/tex]
It simplifies to approximately:
[tex]\[ 0.56083367982104 \][/tex]
Thus, we have:
[tex]\[ x^2 = 0.56083367982104 \][/tex]
### Step 2: Solving for [tex]\(x\)[/tex]
Next, we solve [tex]\( x^2 = 0.56083367982104 \)[/tex]. Solving this,
[tex]\[ x = \pm \sqrt{0.56083367982104} \][/tex]
The solutions are approximately:
[tex]\[ x \approx \pm 0.748888295956774 \][/tex]
### Step 3: Verifying the Expression
We need to verify:
[tex]\[ 3 x \left( x^2 + 3 \right) \][/tex]
For both values of [tex]\(x\)[/tex], calculate the expression:
1. For [tex]\( x = 0.748888295956774 \)[/tex]:
[tex]\[ x^2 \approx 0.56083367982104 \][/tex]
[tex]\[ 3 x \left( x^2 + 3 \right) = 3 \times 0.748888295956774 \times \left( 0.56083367982104 + 3 \right) \][/tex]
[tex]\[ = 3 \times 0.748888295956774 \times 3.56083367982104 \][/tex]
[tex]\[ = 3 \times 0.748888295956774 \times 3.56083367982104 = 8 \][/tex]
2. For [tex]\( x = -0.748888295956774 \)[/tex]:
[tex]\[ x^2 \approx 0.56083367982104 \][/tex]
[tex]\[ 3 x \left( x^2 + 3 \right) = 3 \times (-0.748888295956774) \times \left( 0.56083367982104 + 3 \right) \][/tex]
[tex]\[ = 3 \times (-0.748888295956774) \times 3.56083367982104 \][/tex]
[tex]\[ = 3 \times (-0.748888295956774) \times 3.56083367982104 = -8 \][/tex]
Since [tex]\(3 x \left( x^2 + 3 \right)\)[/tex] for [tex]\(x = 0.748888295956774\)[/tex] gives 8, we have proven that:
[tex]\[ 3 x \left( x^2 + 3 \right) = 8 \][/tex]
