If `((1)/(2))^(y)=(1)/(4)xx2^(y)`, what is y?

Answer

To solve the equation \(\left(\frac{1}{2}\right)^{y} = \frac{1}{4} \times 2^{y}\), we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ \left(\frac{1}{2}\right)^{y} = \frac{1}{4} \times 2^{y} \]

### Step 2: Express \(\frac{1}{2}\) and \(\frac{1}{4}\) in terms of powers of 2
We know that:
\[ \frac{1}{2} = 2^{-1} \quad \text{and} \quad \frac{1}{4} = 2^{-2} \]
Thus, we can rewrite the left side:
\[ \left(2^{-1}\right)^{y} = 2^{-2} \times 2^{y} \]

### Step 3: Simplify the left side
Using the power of a power property \((a^m)^n = a^{m \cdot n}\), we have:
\[ 2^{-y} = 2^{-2} \times 2^{y} \]

### Step 4: Combine the right side
Using the property of exponents \(a^m \times a^n = a^{m+n}\), we combine the terms on the right:
\[ 2^{-y} = 2^{-2 + y} \]

### Step 5: Set the exponents equal
Since the bases are the same, we can set the exponents equal to each other:
\[ -y = -2 + y \]

### Step 6: Solve for \(y\)
To isolate \(y\), we can add \(y\) to both sides:
\[ -y + y = -2 + y + y \implies 0 = -2 + 2y \]
Now, add 2 to both sides:
\[ 2 = 2y \]
Finally, divide both sides by 2:
\[ y = 1 \]

### Final Answer
Thus, the value of \(y\) is:
\[ \boxed{1} \]