If `(4^(1/3))^(2x+1/2)=1/32`, then x =

Options

A. -2
B. 4
C. -6
D. -4

Answer

To solve the equation \((4^{1/3})^{2x + 1/2} = \frac{1}{32}\), we will follow these steps:

### Step 1: Rewrite the left-hand side
We start with the left-hand side of the equation:
\[ (4^{1/3})^{2x + 1/2} \]
We know that \(4\) can be expressed as \(2^2\). Therefore, we can rewrite \(4^{1/3}\) as:
\[ (2^2)^{1/3} = 2^{2/3} \]
Now substituting this back into the equation gives us:
\[ (2^{2/3})^{2x + 1/2} \]

### Step 2: Simplify the left-hand side
Using the power of a power property \((a^m)^n = a^{m \cdot n}\), we can simplify the left-hand side:
\[ 2^{(2/3)(2x + 1/2)} \]
This simplifies to:
\[ 2^{\frac{2(2x + 1/2)}{3}} = 2^{\frac{4x + 1}{3}} \]

### Step 3: Rewrite the right-hand side
Now, we rewrite the right-hand side \(\frac{1}{32}\). Since \(32\) can be expressed as \(2^5\), we have:
\[ \frac{1}{32} = 2^{-5} \]

### Step 4: Set the exponents equal
Now, we can set the exponents equal to each other since the bases are the same:
\[ \frac{4x + 1}{3} = -5 \]

### Step 5: Solve for \(x\)
To eliminate the fraction, we multiply both sides by \(3\):
\[ 4x + 1 = -15 \]
Next, we subtract \(1\) from both sides:
\[ 4x = -16 \]
Finally, we divide both sides by \(4\):
\[ x = -4 \]

### Final Answer
Thus, the value of \(x\) is:
\[ \boxed{-4} \]

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