Solve for x and y if :
`(sqrt(32))^(x)÷2^(y+1)= 1` and `8^(y)-16^(4-(x)/(2))=0`

Answer

To solve the equations given:

1. **Equation 1**: \((\sqrt{32})^x \div 2^{y+1} = 1\)
2. **Equation 2**: \(8^y - 16^{4 - \frac{x}{2}} = 0\)

Let's solve these step by step.

### Step 1: Simplify Equation 1
We start with the first equation:

\[ (\sqrt{32})^x \div 2^{y+1} = 1 \]

We know that \(\sqrt{32} = 32^{1/2}\) and \(32 = 2^5\), so:

\[ \sqrt{32} = (2^5)^{1/2} = 2^{5/2} \]

Thus, we can rewrite the equation as:

\[ (2^{5/2})^x \div 2^{y+1} = 1 \]

This simplifies to:

\[ 2^{(5/2)x} \div 2^{y+1} = 1 \]

Using the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\):

\[ 2^{(5/2)x - (y + 1)} = 1 \]

Since \(2^0 = 1\), we have:

\[ (5/2)x - (y + 1) = 0 \]

This leads to:

\[ (5/2)x - y - 1 = 0 \quad \text{or} \quad 5x - 2y = 2 \quad \text{(Equation 3)} \]

### Step 2: Simplify Equation 2
Now, let's simplify the second equation:

\[ 8^y - 16^{4 - \frac{x}{2}} = 0 \]

We know that \(8 = 2^3\) and \(16 = 2^4\), so we can rewrite the equation:

\[ (2^3)^y - (2^4)^{4 - \frac{x}{2}} = 0 \]

This simplifies to:

\[ 2^{3y} - 2^{4(4 - \frac{x}{2})} = 0 \]

Now, simplifying \(4(4 - \frac{x}{2})\):

\[ 4(4 - \frac{x}{2}) = 16 - 2x \]

Thus, we have:

\[ 2^{3y} - 2^{16 - 2x} = 0 \]

This implies:

\[ 3y = 16 - 2x \quad \text{(Equation 4)} \]

### Step 3: Solve the System of Equations
Now we have two equations:

1. \(5x - 2y = 2\) (Equation 3)
2. \(3y + 2x = 16\) (from rearranging Equation 4)

We can multiply Equation 3 by 2 to eliminate \(y\):

\[ 10x - 4y = 4 \quad \text{(Equation 5)} \]

Now, multiply Equation 4 by 3:

\[ 9y + 6x = 48 \quad \text{(Equation 6)} \]

Now we have:

1. \(10x - 4y = 4\) (Equation 5)
2. \(6x + 9y = 48\) (Equation 6)

### Step 4: Solve for \(y\)
Now let's solve these two equations. We can express \(y\) from Equation 5:

\[ 4y = 10x - 4 \quad \Rightarrow \quad y = \frac{10x - 4}{4} = \frac{5x - 2}{2} \]

Substituting \(y\) in Equation 6:

\[ 6x + 9\left(\frac{5x - 2}{2}\right) = 48 \]

Multiply through by 2 to eliminate the fraction:

\[ 12x + 9(5x - 2) = 96 \]

Expanding:

\[ 12x + 45x - 18 = 96 \]

Combine like terms:

\[ 57x - 18 = 96 \]

Add 18 to both sides:

\[ 57x = 114 \]

Now divide by 57:

\[ x = 2 \]

### Step 5: Solve for \(y\)
Now substitute \(x = 2\) back into Equation 3:

\[ 5(2) - 2y = 2 \]

This simplifies to:

\[ 10 - 2y = 2 \]

Subtract 10 from both sides:

\[ -2y = -8 \]

Dividing by -2:

\[ y = 4 \]

### Final Solution
Thus, the solution for \(x\) and \(y\) is:

\[ x = 2, \quad y = 4 \]