Solve Complex Expression: [(8²-3+5²+7·2)²:100]·(100:10)

We will simplify this expression while maintaining the order of operations which states that parentheses come before multiplication and division,which come before addition and subtraction.

Let's start first by simplifying the expressions in the parentheses, we will note that in this expression there are two pairs of parentheses between which multiplication takes place.

Notice that the left inner parentheses are raised to a power, so let's start simplifying the expression which is within the inner parentheses.

$\big((8^2-3+5^2+7\cdot2)^2:100 \big)\cdot(100:10)=\\ \big((64-3+25+14)^2:100 \big)\cdot(100:10)=\\ \big(100^2:100 \big)\cdot(100:10)$

We simplified the expression which is in the inner parentheses found within the left parentheses.

We did this in two steps because there are addition and subtraction operations between terms in parentheses, and there is also multiplication of terms (according to the order of operations, we first calculated the terms in parentheses, then we calculated the result of the multiplication in these parentheses and then we performed the addition and subtraction operations which are in the parentheses).

Then, we will simplify the expression which is in the left parentheses first, and only then we will simplify the expression which is in the right parentheses.

We will start by calculating the term in parentheses since parentheses precede multiplication and division, then we will perform the division operation which is in the parentheses:

$\big(100^2:100 \big)\cdot(100:10)=\\ \big(10000:100 \big)\cdot(100:10)=\\ 100\cdot(100:10)=\\ 100\cdot10=\\ 1000$In the last steps we divided within the right set of parentheses and finally we multiplied.

Let's summarize the steps of simplifying the given expression:

$\big((8^2-3+5^2+7\cdot2)^2:100 \big)\cdot(100:10)=\\ \big(100^2:100 \big)\cdot(100:10)=\\ 100\cdot(100:10)=\\ 100\cdot10=\\ 1000$

Therefore the correct answer is answer C.

Note:

The expression in the left parentheses in the last steps can be calculated numerically step by step as described there, but note that it is also possible to reach the same result without calculating their numerical value of the terms in the expression, by using the law of exponents to give terms with identical bases:

$\frac{a^m}{a^n}=a^{m-n}$

This is done as follows:
$100^2:100=\\ \frac{100^2}{100}=\\ 100^{2-1}=\\ 100$

First we converted the division operation to a fraction, then we applied the above law of exponents while remembering that any number can be represented as the same number to the power of 1 (and any number to the power of 1 equals the number itself) .