91Ó°ÊÓ

When we talk about negative exponents, it's crucial to remember that they represent the reciprocal of the base raised to the opposite positive exponent. For instance, \(x^{-n} = \frac{1}{x^n}\), where \(x\) is the base and \(n\) is a positive integer. When dealing with negative exponents, you should first flip the base's position (from numerator to denominator or vice versa) and then apply the positive exponent. In practice, however, you won't encounter a negative exponent every single time you have a negative sign; be mindful of where the negative sign is placed. If it's inside parentheses with the base, as in the given exercise, it affects the base itself. Remember, a negative base raised to an odd power will result in a negative outcome, just like \( (-3)^3 = -27 \) from our given solution.

Simplifying expressions is like tidying up a room; you're organizing and reducing the equation to its simplest form to make it easier to understand and solve. Begin by applying any exponents, then multiply or divide from left to right, and, if applicable, wrap up by adding or subtracting. With our exercise, \(5(-3)^3\), the simplification process involved handling the exponent first, which gave us \(5 * -27\). Then, we multiplied the 5 and -27 to get the final, simplified answer of \-135\. The main goal is to combine like terms and use the order of operations properly to ensure the expression is as simple as it can be.

Substitution is a method used in algebra to solve equations and evaluate expressions. It involves replacing a variable with a given number or another expression. Like in a recipe, you swap out ingredients based on what you have – it's a direct replacement. In our exercise, we were asked to evaluate \(5(-x)^3\) for \(x = 3\). By substituting 3 for \(x\), the way you would swap sugar for honey in a recipe, the expression transformed into \(5(-3)^3\). Substitution is essential for understanding and working through algebraic expressions and equations as it allows you to see what happens to the entire expression when you change just one part.