If `a-b= 0.9` and ab= 0.36 find:
`a^(2) -b^(2)`

Answer

To solve the problem step by step, we need to find the value of \( a^2 - b^2 \) given that \( a - b = 0.9 \) and \( ab = 0.36 \).

### Step 1: Use the identity for \( a^2 - b^2 \)
We know that:
\[ a^2 - b^2 = (a - b)(a + b) \]
From the problem, we have \( a - b = 0.9 \).

### Step 2: Find \( a + b \)
To find \( a + b \), we can use the identity:
\[ (a - b)^2 = a^2 + b^2 - 2ab \]
Substituting the known values:
\[ (0.9)^2 = a^2 + b^2 - 2(0.36) \]
Calculating \( (0.9)^2 \):
\[ 0.81 = a^2 + b^2 - 0.72 \]
Now, rearranging the equation to find \( a^2 + b^2 \):
\[ a^2 + b^2 = 0.81 + 0.72 = 1.53 \]

### Step 3: Use the identity for \( a + b \)
We can also use the identity:
\[ (a + b)^2 = a^2 + b^2 + 2ab \]
Substituting the values we have:
\[ (a + b)^2 = 1.53 + 2(0.36) \]
Calculating \( 2(0.36) \):
\[ 2(0.36) = 0.72 \]
So,
\[ (a + b)^2 = 1.53 + 0.72 = 2.25 \]
Taking the square root to find \( a + b \):
\[ a + b = \sqrt{2.25} = 1.5 \]

### Step 4: Substitute back to find \( a^2 - b^2 \)
Now we have:
- \( a - b = 0.9 \)
- \( a + b = 1.5 \)

Now we can find \( a^2 - b^2 \):
\[ a^2 - b^2 = (a - b)(a + b) = (0.9)(1.5) \]
Calculating this:
\[ a^2 - b^2 = 1.35 \]

### Final Answer
Thus, the value of \( a^2 - b^2 \) is:
\[ \boxed{1.35} \]