### Instruction: Evaluate the expression \[ (a^2+b)^2 - (a^2-b)^2, \]if $a=4$ and $b=1$.### Response: To evaluate this expression, I need to plug in the values of $a$ and $b$ and simplify using the order of operations.
First, I replace $a$ with 4 and $b$ with 1 in the expression: \[ (4^2+1)^2 - (4^2-1)^2 \]
Next, I simplify the exponents inside the parentheses: \[ (16+1)^2 - (16-1)^2 \]
Then, I simplify the additions and subtractions inside the parentheses: \[ (17)^2 - (15)^2 \]
Now, I evaluate the squares of 17 and 15: \[ 289 - 225 \]
Finally, I subtract 225 from 289: \[ 64 \]