\text{The value of}{\mathrm{ 4}}^3-7^2+{\left(\frac{16}{4}\right)}^2-4+9\times 1=\text{\_\_\_\_\_\_}

The correct option is 36.

To find the value of the given mathematical expression, we will simplify each term step-by-step following the standard order of operations (PEMDAS/BODMAS).

The given expression is:

$$4^3-7^2+{\left(\frac{16}{4}\right)}^2-4+9\times 1$$

Step 1: Simplify terms inside parentheses
We first calculate the division inside the parentheses:
$\frac{16}{4}=4$
Substituting this back, the expression becomes:
$4^3-7^2+4^2-4+9\times 1$

Step 2: Evaluate the exponents (powers)
Now we calculate the value of each exponential term:
1. First term: 4³ = 4 × 4 × 4 = 64
2. Second term: 7² = 7 × 7 = 49
3. Third term: 4² = 4 × 4 = 16
Substituting these values back into the expression, we get:
$64-49+16-4+9\times 1$

Step 3: Perform multiplication
Next, evaluate the multiplication at the end of the expression:
9 × 1 = 9
So, the expression becomes:
$64-49+16-4+9$

Step 4: Perform addition and subtraction from left to right
Let us perform the calculations sequentially from left to right:
1. 64 - 49 = 15
The expression now is: 15 + 16 - 4 + 9
2. 15 + 16 = 31
The expression now is: 31 - 4 + 9
3. 31 - 4 = 27
The expression now is: 27 + 9
4. 27 + 9 = 36

Thus, the final value of the expression is 36.