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Understanding the order of operations is crucial when solving algebraic expressions. It is often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). In some regions, it's also known as BIDMAS or BODMAS, where 'B' stands for Brackets and 'O' for Orders (powers and root operations).

Following this rule ensures that mathematical expressions are solved consistently and accurately. For the given exercise, \(\frac{4(-3)^{2}}{-2}\), we start with the operation inside the parentheses, which is squaring -3. This adheres to the 'Exponents' part of our order of operations. Once the exponent is evaluated, we move on to multiplication, and finally, division. Applying these rules prevents confusion and ensures that anyone working out the problem will reach the same solution if they follow PEMDAS/BIDMAS accurately.

Exponents and powers represent the repeated multiplication of a number by itself. An exponent is noted by a small number written to the upper right of the base number. For example, in the expression \( (-3)^{2} \), -3 is the base and 2 is the exponent, which means -3 should be multiplied by itself once (since squaring is the same as raising to the power of 2).

It is crucial to note that the exponent applies only to the number it is directly above when no parentheses are present. However, when parentheses are included, as in our example, the exponent applies to the entire contents within. Therefore, \( (-3)^{2} \), means we must square -3, not 3, resulting in 9. Remember that squaring a negative number gives a positive result because a negative multiplied by a negative equals a positive.

Arithmetic operations within algebraic contexts include the familiar operations: addition, subtraction, multiplication, and division, yet they work with numerals, unknowns (variables), and the combination of both.

In the given expression, \(\frac{4(-3)^{2}}{-2}\), we encounter a multiplication and a division operation. Multiplication in algebra follows the same rules as in standard arithmetic. If you are multiplying numbers with the same sign, the result is positive; if the numbers have different signs, the result is negative. Afterwards, division is handled, and as with multiplication, the signs are important. Dividing two numbers with the same sign gives a positive result, while dividing with different signs produces a negative outcome.

The exercise showcases these arithmetic operations within an algebraic expression, providing a clear example of how numbers and operations interact in algebra.