Powers and Roots: Using 0

To solve the expression, we need to follow the order of operations, often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (left-to-right), Addition and Subtraction (left-to-right).

• Step 1: Handle the exponentiation in the expression.
  The exponentiation operation is on the number 2, so we compute $2^2$ which is equal to 4.
• Step 2: Handle the multiplication.
  We multiply the result of the exponentiation by the number following it. Therefore, we have $4 \times 1^{10}$. Since $1^{10} = 1$, the expression becomes $4 \times 1 = 4$.
• Step 3: Handle the operations on either side of multiplication.
  Before evaluating, it's essential to note that the original expression is slightly misunderstood due to potential formatting errors. Let's assume it was meant to use division as $0 \div 2^2$ due to the notation $0:2^2$. Therefore, calculate $0 \div 4$ which is equal to 0.
• Step 4: Handle the addition.
  Finally, we add 3 to the result of the division: $0 + 3 = 3$.

Thus, the final answer is 3.

Let's solve the problem step by step using the Zero Exponent Rule, which states that any non-zero number raised to the power of 0 is equal to 1.

• Consider the expression: $100^0$.
• According to the Zero Exponent Rule, if we have any non-zero number, say $a$, then $a^0 = 1$.
• Here, $a = 100$ which is clearly a non-zero number, so following the rule, we find that:
• $100^0 = 1$.

Therefore, the expression $100^0$ is equivalent to 1.

This simple example demonstrates the order of operations, which states that exponentiation precedes multiplication and division, which come before addition and subtraction, and that operations within parentheses come first,

In the given example, the operation of division between parentheses (the denominators) by a number (which is also in parentheses but only for clarification purposes), thus according to the order of operations mentioned we start with the parentheses that contain the denominators first, this parentheses that contain the denominators includes multiplication between two numbers which are also in parentheses, therefore according to the order of operations mentioned, we start with the numbers inside them, paying attention that each of these numbers, including the ones in strength, and therefore assuming that exponentiation precedes multiplication and division we consider their numerical values only in the first step and only then do we perform the operations of multiplication and division on these numbers:

$$$$$\lbrack(3^2-4-5)\cdot(4+\sqrt{16})-5 \rbrack:(-5)=\\ \lbrack(9-4-5)\cdot(4+4)-5 \rbrack:(-5)=\\ \lbrack0\cdot8-5 \rbrack:(-5)\\$Continuing with the simple division in parentheses ,and according to the order of operations mentioned, we proceed from the multiplication calculation and remember that the multiplication of the number 0 by any number will yield the result 0, in the first step the operation of subtraction is performed and finally the operation of division is initiated on the number in parentheses:

$\lbrack0\cdot8-5 \rbrack:(-5)= \\ \lbrack0-5 \rbrack:(-5)= \\ -5 :(-5)=\\ 1$ Therefore, the correct answer is answer c.