If `f(x) = 2x^4 - x^2`, what is the value of `f(2sqrt(3))`?

Answer

To find the value of \( f(2\sqrt{3}) \) for the function \( f(x) = 2x^4 - x^2 \), we will follow these steps:

### Step 1: Substitute \( 2\sqrt{3} \) into the function
We start by substituting \( x = 2\sqrt{3} \) into the function:
\[ f(2\sqrt{3}) = 2(2\sqrt{3})^4 - (2\sqrt{3})^2 \]

### Step 2: Calculate \( (2\sqrt{3})^4 \)
Now we calculate \( (2\sqrt{3})^4 \):
\[ (2\sqrt{3})^4 = (2^4)(\sqrt{3})^4 = 16 \cdot 9 = 144 \]

### Step 3: Calculate \( (2\sqrt{3})^2 \)
Next, we calculate \( (2\sqrt{3})^2 \):
\[ (2\sqrt{3})^2 = 2^2 \cdot (\sqrt{3})^2 = 4 \cdot 3 = 12 \]

### Step 4: Substitute back into the function
Now we substitute these values back into the function:
\[ f(2\sqrt{3}) = 2 \cdot 144 - 12 \]

### Step 5: Perform the multiplication
Calculate \( 2 \cdot 144 \):
\[ 2 \cdot 144 = 288 \]

### Step 6: Final calculation
Now, we subtract \( 12 \) from \( 288 \):
\[ 288 - 12 = 276 \]

### Conclusion
Thus, the value of \( f(2\sqrt{3}) \) is:
\[ \boxed{276} \]

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