Determine the approximate value that should replace the question mark (?) in the following mathematical expression:

$2^{?}\%\text{of}499.99+\sqrt{1296.01}={\left(13.98\right)}^2$

To find the approximate value that should replace the question mark (?) in the given expression, let us first state the correct answer clearly:

The correct answer is 5.

Now, let us simplify the given mathematical expression step-by-step using approximation.

The original equation is:

$2^{?}\%\text{of}499.99+\sqrt{1296.01}={\left(13.98\right)}^2$

Step 1: Approximate the decimal values to the nearest whole numbers.

1. $499.99\approx 500$

2. $1296.01\approx 1296$

3. $13.98\approx 14$

Step 2: Substitute these approximate values back into the equation.

$2^{?}\%\text{of}500+\sqrt{1296}={14}^2$

Step 3: Calculate the known mathematical terms.

Find the square root of 1296:

$\sqrt{1296}=36$

Calculate the square of 14:

${14}^2=196$

Step 4: Rewrite the equation with the evaluated values.

$\left(\frac{2^{?}}{100}\right)\times 500+36=196$

Simplifying $\frac{500}{100}$ gives 5:

$2^{?}\times 5+36=196$

Step 5: Solve for the exponent term ($2^{?}$).

Subtract 36 from both sides of the equation:

$2^{?}\times 5=196-36$

$2^{?}\times 5=160$

Now, divide both sides by 5:

$2^{?}=\frac{160}{5}$

$2^{?}=32$

Step 6: Express 32 as a power of 2 to find the value of (?).

We know that:

$32=2^5$

Equating the exponents on both sides:

$?=5$

Thus, the approximate value that replaces the question mark is 5.