Evaluate : `((1)/(2))^(-4) xx ((1)/(2))^(-8) xx ((1)/(2))^(2) + ((1)/(4))^(2) xx ((1)/(4) )^(-6) xx ((1)/(4))^(2)`

Options

A. `((1)/(2))^(-10)`
B. `((1)/(2))^(12) xx ((1)/(4))^(-1)`
C. `((1)/(2))^(-14) + ((1)/(4))^(0)`
D. `((1)/(2))^(12)`

Answer

To evaluate the expression \(\left(\frac{1}{2}\right)^{-4} \times \left(\frac{1}{2}\right)^{-8} \times \left(\frac{1}{2}\right)^{2} + \left(\frac{1}{4}\right)^{2} \times \left(\frac{1}{4}\right)^{-6} \times \left(\frac{1}{4}\right)^{2}\), we can follow these steps:

### Step 1: Simplify the first part of the expression
We start with the first part of the expression:
\[ \left(\frac{1}{2}\right)^{-4} \times \left(\frac{1}{2}\right)^{-8} \times \left(\frac{1}{2}\right)^{2} \]
Since the bases are the same, we can add the exponents:
\[ = \left(\frac{1}{2}\right)^{-4 + (-8) + 2} \]
Calculating the exponent:
\[ -4 - 8 + 2 = -10 \]
Thus, we have:
\[ = \left(\frac{1}{2}\right)^{-10} \]

### Step 2: Simplify the second part of the expression
Now, we simplify the second part:
\[ \left(\frac{1}{4}\right)^{2} \times \left(\frac{1}{4}\right)^{-6} \times \left(\frac{1}{4}\right)^{2} \]
Again, since the bases are the same, we can add the exponents:
\[ = \left(\frac{1}{4}\right)^{2 + (-6) + 2} \]
Calculating the exponent:
\[ 2 - 6 + 2 = -2 \]
Thus, we have:
\[ = \left(\frac{1}{4}\right)^{-2} \]

### Step 3: Combine the results
Now we combine both parts:
\[ \left(\frac{1}{2}\right)^{-10} + \left(\frac{1}{4}\right)^{-2} \]

### Step 4: Convert negative exponents to positive
Using the property \(a^{-b} = \frac{1}{a^b}\):
\[ \left(\frac{1}{2}\right)^{-10} = 2^{10} \]
\[ \left(\frac{1}{4}\right)^{-2} = 4^{2} \]

### Step 5: Calculate the powers
Calculating \(2^{10}\) and \(4^{2}\):
\[ 2^{10} = 1024 \]
\[ 4^{2} = 16 \]

### Step 6: Add the results
Now we add the two results:
\[ 1024 + 16 = 1040 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{1040} \]