Determine the approximate value that should replace the question mark (?) in the following mathematical equation. (You do not need to calculate the exact value.)

$\sqrt{1764.02}+{21.04}^2-?=\frac{5}{12}\times 167.97$

The correct option is 413.

To find the approximate value that replaces the question mark (?), let us round each term in the given equation to its nearest integer or easily solvable value.

The given equation is:

$\sqrt{1764.02}+{21.04}^2-?=\frac{5}{12}\times 167.97$

Step 1: Approximate each term

1. $\sqrt{1764.02}\approx \sqrt{1764}=42$ (since 42² = 1764)

2. ${21.04}^2\approx {21}^2=441$

3. $\frac{5}{12}\times 167.97\approx \frac{5}{12}\times 168=5\times 14=70$

Step 2: Substitute the approximate values back into the equation

$42+441-?=70$

Step 3: Solve for ?

$483-?=70$

$?=483-70$

$?=413$

Therefore, the approximate value that replaces the question mark is 413.