If `2^(2y)=(1)/(4) and 3^(4x)=(1)/(81)` then x+y ____

Options

A. 0
B. `-2`
C. 1
D. 2

Answer

To solve the equations given in the problem, we will follow these steps:

### Step 1: Solve for y from the first equation
We start with the equation:
\[ 2^{2y} = \frac{1}{4} \]

We can express \( \frac{1}{4} \) as a power of 2:
\[ \frac{1}{4} = 2^{-2} \]

Now we have:
\[ 2^{2y} = 2^{-2} \]

Since the bases are the same, we can set the exponents equal to each other:
\[ 2y = -2 \]

Now, divide both sides by 2 to solve for y:
\[ y = \frac{-2}{2} = -1 \]

### Step 2: Solve for x from the second equation
Now we consider the second equation:
\[ 3^{4x} = \frac{1}{81} \]

We can express \( \frac{1}{81} \) as a power of 3:
\[ \frac{1}{81} = 3^{-4} \]

Now we have:
\[ 3^{4x} = 3^{-4} \]

Again, since the bases are the same, we can set the exponents equal:
\[ 4x = -4 \]

Now, divide both sides by 4 to solve for x:
\[ x = \frac{-4}{4} = -1 \]

### Step 3: Calculate x + y
Now that we have the values of x and y:
\[ x = -1 \]
\[ y = -1 \]

We can find \( x + y \):
\[ x + y = -1 + (-1) = -2 \]

### Final Answer
Thus, the value of \( x + y \) is:
\[ \boxed{-2} \]

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