Estimate the value that should replace the question mark (?) in the following expression:

$109.09+511.98÷15.97-?{\mathrm{²}}^{}=\left(20.99\right)2^{}-($\(19.96\) ) 2

The correct answer is 10.

We need to estimate the value that should replace the question mark (?) in the following expression:

$109.09+511.98÷15.97-{?}^2={\left(20.99\right)}^2-{\left(19.96\right)}^2$

Step 1: Round off the numbers to their nearest whole numbers for estimation.

109.09 ≈ 109
511.98 ≈ 512
15.97 ≈ 16
20.99 ≈ 21
19.96 ≈ 20

Step 2: Substitute the approximated values into the given equation.

$109+512÷16-{?}^2={21}^2-{20}^2$

Step 3: Perform division according to the BODMAS rule.

$512÷16=32$

Substituting this result back into the expression:

$109+32-{?}^2={21}^2-{20}^2$

Step 4: Simplify the left-hand side and right-hand side.

Left-hand side addition:

$109+32=141$

Right-hand side squares calculation:

${21}^2=441$

${20}^2=400$

$441-400=41$

Step 5: Solve for the unknown value (?).

Now, rewrite the simplified equation:

$141-{?}^2=41$

${?}^2=141-41$

${?}^2=100$

$?=\sqrt{100}=10$

Therefore, the value that replaces the question mark (?) is 10.