If `((-(1)/(2))^(6))/((-(1)/(2))^(4))-:(((1)/(4))^(13))/(((1)/(4))^(10))=2^(x)`, then find the value of x.

Options

A. 4
B. 0
C. 1
D. `-4`

Answer

To solve the equation

\[ \frac{(-\frac{1}{2})^6}{(-\frac{1}{2})^4} \div \frac{(\frac{1}{4})^{13}}{(\frac{1}{4})^{10}} = 2^x, \]

we will follow these steps:

### Step 1: Simplify the left-hand side using the properties of exponents.
Using the property of exponents that states \(\frac{a^m}{a^n} = a^{m-n}\), we can simplify both fractions.

1. For the first part:
\[ \frac{(-\frac{1}{2})^6}{(-\frac{1}{2})^4} = (-\frac{1}{2})^{6-4} = (-\frac{1}{2})^2. \]

2. For the second part:
\[ \frac{(\frac{1}{4})^{13}}{(\frac{1}{4})^{10}} = (\frac{1}{4})^{13-10} = (\frac{1}{4})^3. \]

### Step 2: Rewrite the expression.
Now, we can rewrite the equation as:
\[ \frac{(-\frac{1}{2})^2}{(\frac{1}{4})^3}. \]

### Step 3: Calculate \((- \frac{1}{2})^2\) and \((\frac{1}{4})^3\).
1. Calculate \((- \frac{1}{2})^2\):
\[ (-\frac{1}{2})^2 = \frac{1}{4}. \]

2. Calculate \((\frac{1}{4})^3\):
\[ (\frac{1}{4})^3 = \frac{1}{64}. \]

### Step 4: Substitute back into the equation.
Now substituting these values back into our equation gives:
\[ \frac{\frac{1}{4}}{\frac{1}{64}}. \]

### Step 5: Simplify the division.
Dividing \(\frac{1}{4}\) by \(\frac{1}{64}\) is the same as multiplying by the reciprocal:
\[ \frac{1}{4} \times 64 = \frac{64}{4} = 16. \]

### Step 6: Rewrite 16 in terms of base 2.
Now we can express 16 as a power of 2:
\[ 16 = 2^4. \]

### Step 7: Set the equation equal to \(2^x\).
Now we have:
\[ 2^4 = 2^x. \]

### Step 8: Solve for \(x\).
Since the bases are the same, we can equate the exponents:
\[ x = 4. \]

### Final Answer:
The value of \(x\) is \(4\).

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