Estimate the value that should replace the question mark (?) in the mathematical expression below.

$?\times (20.01\%\text{of}580.01+\sqrt{196.01})={\left(}^{8.99}+17.01\times 3.03$

The correct option is 6.

To estimate the value that replaces the question mark (?), we can round each decimal number in the expression to its nearest integer for ease of calculation.

Let's analyze each component of the given equation:

$?\times (20.01\%\text{of}580.01+\sqrt{196.01})={\left(8.99\right)}^3+17.01\times 3.03$

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

• 20.01% ≈ 20%
• 580.01 ≈ 580
• 196.01 ≈ 196
• 8.99 ≈ 9
• 17.01 ≈ 17
• 3.03 ≈ 3

Step 2: Simplify the expression inside the parentheses on the Left-Hand Side (LHS).

First, calculate 20% of 580:

$20\%\text{of}580=\frac{20}{100}\times 580=116$

Next, find the square root of 196:

$\sqrt{196}=14$

Now, add these two values together:

$116+14=130$

So, the LHS simplifies to: $?\times 130$

Step 3: Simplify the Right-Hand Side (RHS).

Calculate the cube of 9:

$9^3=729$

Calculate the product of 17 and 3:

$17\times 3=51$

Add these two values together:

$729+51=780$

Step 4: Solve for the question mark (?).

Equate LHS to RHS:

$?\times 130=780$

$?=\frac{780}{130}$

$?=6$

Thus, the estimated value that replaces the question mark is 6.