Find the value of `a^(2)+b^(2)` if `a+b=10anda-b=2`.

Answer

To find the value of \( a^2 + b^2 \) given the equations \( a + b = 10 \) and \( a - b = 2 \), we can follow these steps:

### Step 1: Square both equations
We start with the two equations:
1. \( a + b = 10 \)
2. \( a - b = 2 \)

Now, we square both equations.

- For the first equation:
\[ (a + b)^2 = 10^2 \]
This expands to:
\[ a^2 + 2ab + b^2 = 100 \quad \text{(Equation 1)} \]

- For the second equation:
\[ (a - b)^2 = 2^2 \]
This expands to:
\[ a^2 - 2ab + b^2 = 4 \quad \text{(Equation 2)} \]

### Step 2: Add the two equations
Now, we will add Equation 1 and Equation 2:
\[ (a^2 + 2ab + b^2) + (a^2 - 2ab + b^2) = 100 + 4 \]

### Step 3: Simplify the left side
On the left side, the \( 2ab \) and \( -2ab \) cancel each other out:
\[ a^2 + a^2 + b^2 + b^2 = 104 \]
This simplifies to:
\[ 2a^2 + 2b^2 = 104 \]

### Step 4: Divide by 2
Next, we divide the entire equation by 2 to isolate \( a^2 + b^2 \):
\[ a^2 + b^2 = \frac{104}{2} \]
This gives us:
\[ a^2 + b^2 = 52 \]

### Final Answer
Thus, the value of \( a^2 + b^2 \) is \( \boxed{52} \).

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