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 answer is 413.

To find the approximate value of the question mark (?), we round off each decimal term in the given equation to its nearest whole number or convenient value.

The given equation is:

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

Step 1: Approximate each term of the equation.

1. Approximate $\sqrt{1764.02}$:

$\sqrt{1764.02}\approx \sqrt{1764}=42$

2. Approximate ${21.04}^2$:

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

3. Approximate $\frac{5}{12}\times 167.97$:

$\frac{5}{12}\times 167.97\approx \frac{5}{12}\times 168$

Dividing 168 by 12 gives 14:

$5\times 14=70$

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

$42+441-?=70$

$483-?=70$

Step 3: Solve for the question mark (?).

$?=483-70$

$?=413$

Therefore, the approximate value that should replace the question mark is 413.