If `y=2^(2p-1) and z=p-2,` what is the value of `(y)/(z)` when `p=2.5`?

Answer

To solve the problem, we need to find the value of \(\frac{y}{z}\) when \(p = 2.5\). We are given the equations:

1. \(y = 2^{(2p - 1)}\)
2. \(z = p - 2\)

### Step 1: Substitute \(p = 2.5\) into the equation for \(y\)

\[ y = 2^{(2p - 1)} = 2^{(2 \times 2.5 - 1)} \]

### Step 2: Calculate \(2 \times 2.5\)

\[ 2 \times 2.5 = 5 \]

### Step 3: Substitute back into the equation for \(y\)

\[ y = 2^{(5 - 1)} = 2^{4} \]

### Step 4: Calculate \(2^{4}\)

\[ 2^{4} = 16 \]

### Step 5: Substitute \(p = 2.5\) into the equation for \(z\)

\[ z = p - 2 = 2.5 - 2 \]

### Step 6: Calculate \(2.5 - 2\)

\[ 2.5 - 2 = 0.5 \]

### Step 7: Calculate \(\frac{y}{z}\)

\[ \frac{y}{z} = \frac{16}{0.5} \]

### Step 8: Simplify \(\frac{16}{0.5}\)

To simplify \(\frac{16}{0.5}\), we can multiply both the numerator and the denominator by 2:

\[ \frac{16 \times 2}{0.5 \times 2} = \frac{32}{1} = 32 \]

### Final Answer

The value of \(\frac{y}{z}\) when \(p = 2.5\) is \(32\).

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