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

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

The correct option is 5.

Let us solve the given mathematical expression step-by-step using approximation.

The given equation is:

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

Let the unknown value represented by the question mark (?) be x.

Step 1: Approximate the numbers to their nearest integers
- 499.99 can be approximated to 500.
- 1296.01 can be approximated to 1296.
- 13.98 can be approximated to 14.

Substituting these approximate values into the equation, we get:

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

Step 2: Calculate the square root and the square
- We know that $\sqrt{1296}=36$
- We also know that ${14}^2=196$

Substitute these calculated values back into the simplified equation:

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

Step 3: Simplify the left-hand side
Since 500 divided by 100 is 5, we have:

$2^x\times 5+36=196$

Subtract 36 from both sides of the equation:

$2^x\times 5=196-36$

$2^x\times 5=160$

Divide both sides by 5:

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

$2^x=32$

Step 4: Solve for x
We can write 32 as a power of 2:

$32=2^5$

Therefore:

$2^x=2^5$

$x=5$

Hence, the approximate value that should replace the question mark (?) is 5.