Solve for a and b :
`2^(a) + 3^(b) = 17` and `2^(a+2) - 3^(b+1) = 5`

Answer

To solve the equations \(2^a + 3^b = 17\) and \(2^{a+2} - 3^{b+1} = 5\), we will follow these steps:

### Step 1: Rewrite the equations
We start with the two equations:
1. \(2^a + 3^b = 17\) (Equation 1)
2. \(2^{a+2} - 3^{b+1} = 5\) (Equation 2)

### Step 2: Simplify Equation 2
We can rewrite Equation 2 as:
\[ 2^{a+2} = 4 \cdot 2^a \]
Thus, Equation 2 becomes:
\[ 4 \cdot 2^a - 3 \cdot 3^b = 5 \]
Rearranging gives us:
\[ 4 \cdot 2^a - 3 \cdot 3^b = 5 \quad \text{(Equation 2')} \]

### Step 3: Substitute \(2^a\) and \(3^b\)
Let:
\[ x = 2^a \quad \text{and} \quad y = 3^b \]
Now we can rewrite our equations:
1. \(x + y = 17\) (Equation 3)
2. \(4x - 3y = 5\) (Equation 4)

### Step 4: Solve the system of equations
We can solve these two equations simultaneously. From Equation 3, we can express \(y\) in terms of \(x\):
\[ y = 17 - x \]

Now substitute \(y\) in Equation 4:
\[ 4x - 3(17 - x) = 5 \]
Expanding this gives:
\[ 4x - 51 + 3x = 5 \]
Combining like terms:
\[ 7x - 51 = 5 \]
Adding 51 to both sides:
\[ 7x = 56 \]
Dividing by 7:
\[ x = 8 \]

### Step 5: Find \(y\)
Now substitute \(x = 8\) back into Equation 3 to find \(y\):
\[ 8 + y = 17 \]
Thus,
\[ y = 17 - 8 = 9 \]

### Step 6: Solve for \(a\) and \(b\)
Now we have:
\[ x = 2^a = 8 \quad \Rightarrow \quad 2^a = 2^3 \quad \Rightarrow \quad a = 3 \]
\[ y = 3^b = 9 \quad \Rightarrow \quad 3^b = 3^2 \quad \Rightarrow \quad b = 2 \]

### Final Answer
Thus, the solution is:
\[ a = 3, \quad b = 2 \]