What is the value of x, if `(-(1)/(2))^(4)xx(-2)^(8)=(-2)^(4x)`?

Answer

To solve the equation \((- \frac{1}{2})^{4} \cdot (-2)^{8} = (-2)^{4x}\), we will follow these steps:

### Step 1: Simplify the left-hand side
We start with the expression \((- \frac{1}{2})^{4}\). We can rewrite it as:
\[ (- \frac{1}{2})^{4} = \frac{(-1)^{4}}{2^{4}} = \frac{1}{16} \]
because \((-1)^{4} = 1\).

### Step 2: Calculate \((-2)^{8}\)
Next, we calculate \((-2)^{8}\):
\[ (-2)^{8} = 256 \]
since raising a negative number to an even power results in a positive number.

### Step 3: Combine the results
Now we can combine the results from Step 1 and Step 2:
\[ (- \frac{1}{2})^{4} \cdot (-2)^{8} = \frac{1}{16} \cdot 256 \]
Calculating this gives:
\[ \frac{256}{16} = 16 \]

### Step 4: Set the left-hand side equal to the right-hand side
Now we have:
\[ 16 = (-2)^{4x} \]

### Step 5: Rewrite 16 as a power of -2
We can express 16 as a power of -2:
\[ 16 = (-2)^{4} \]
Thus, we can rewrite our equation as:
\[ (-2)^{4} = (-2)^{4x} \]

### Step 6: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal to each other:
\[ 4 = 4x \]

### Step 7: Solve for x
Now, we solve for \(x\):
\[ x = \frac{4}{4} = 1 \]

### Final Answer
The value of \(x\) is \(1\).

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