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Understanding how to multiply fractions with integers is a foundational skill in mathematics. In the context of our example \(\frac{2}{3}(-36)\), the process involves a few straightforward steps.

First, you identify the numbers that will be involved in the multiplication: a fraction, \(\frac{2}{3}\), and an integer, -36. The rule for multiplying a fraction by an integer is to multiply the integer by the numerator (top number) of the fraction. Here, you multiply -36 by 2 to get -72.

Finally, you need to divide this product by the denominator (bottom number) of the fraction, which in this case is 3. So, \(\frac{-72}{3}\) simplifies to -24. It's similar to the 'times tables' but with an extra step of dividing by the denominator after multiplying. To avoid common mistakes, remember to perform the operations in the correct order and watch out for negative signs.

When dealing with negative numbers in multiplication, the rules are simple yet crucial for correct answers. The general rule is: multiplying two numbers with the same sign (either both positive or both negative) results in a positive product, while multiplying numbers with opposite signs results in a negative product.

Using our example \(\frac{2}{3}(-36)\), the fraction \(\frac{2}{3}\) is positive, and -36 is negative, so their product must be negative. This is why when we multiply -36 by the numerator 2, we obtain -72, a negative number. Misinterpreting the sign can lead to an incorrect answer, so it's always critical to pay attention to whether numbers are positive or negative when performing multiplication.

Multiplying fractions with integers, as seen in the example \(\frac{2}{3}(-36)\), is an algebraic operation. Algebra involves working with numbers and variables to solve equations and perform calculations.

In our example, although there's no variable, the operation still adheres to algebraic principles such as the distributive property. This means if you had a fraction multiplied by a sum inside parentheses, you would apply the fraction to each term within the parentheses. Here, we only have one number, so it's a straightforward application, but the principle holds true for more complex expressions as well.

Remember that algebraic operations require a systematic approach; first perform multiplications and divisions from left to right, and then additions and subtractions in the same manner. Keeping this order of operations in mind can help you avoid errors in more intricate algebraic calculations.