If `12^(4+2x^(2)) = (24sqrt(3))^(3x^(2)-2)`, then x is equal to

Options

A. `+- sqrt((13)/(12))`
B. `+- sqrt((14)/(5))`
C. `+- ((12)/(13))`
D. `+- sqrt((5)/(14))`

Explanation

To solve the equation \( 12^{4 + 2x^2} = (24\sqrt{3})^{3x^2 - 2} \), we will follow these steps:

### Step 1: Rewrite the equation
Start by rewriting the right-hand side:
\[ 24\sqrt{3} = 12 \cdot 2 \cdot 3^{1/2} \]
Thus, we can express the equation as:
\[ 12^{4 + 2x^2} = (12 \cdot 2 \cdot 3^{1/2})^{3x^2 - 2} \]

### Step 2: Simplify the right-hand side
Using the properties of exponents, we can expand the right-hand side:
\[ (12 \cdot 2 \cdot 3^{1/2})^{3x^2 - 2} = 12^{3x^2 - 2} \cdot 2^{3x^2 - 2} \cdot (3^{1/2})^{3x^2 - 2} \]
This simplifies to:
\[ 12^{3x^2 - 2} \cdot 2^{3x^2 - 2} \cdot 3^{(3x^2 - 2)/2} \]

### Step 3: Equate the bases
Now, we can equate the powers of 12 from both sides:
\[ 12^{4 + 2x^2} = 12^{3x^2 - 2} \cdot 2^{3x^2 - 2} \cdot 3^{(3x^2 - 2)/2} \]
This gives us:
\[ 4 + 2x^2 = 3x^2 - 2 + \text{(additional terms)} \]

### Step 4: Isolate the powers of 12
To isolate the powers of \(12\), we can subtract \(3x^2 - 2\) from both sides:
\[ 4 + 2x^2 + 2 = 3x^2 \]
This simplifies to:
\[ 6 + 2x^2 = 3x^2 \]

### Step 5: Rearrange the equation
Rearranging gives:
\[ 6 = 3x^2 - 2x^2 \]
Thus:
\[ 6 = x^2 \]

### Step 6: Solve for \(x\)
Taking the square root of both sides, we find:
\[ x = \pm \sqrt{6} \]

### Final Answer
The values of \(x\) are:
\[ x = \sqrt{6} \quad \text{or} \quad x = -\sqrt{6} \]

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