Evaluate \(3 x(x-2 a)^{-x}\), if \(x=-2\) and \(a=0.5\).

When working on math problems, substituting values is a technique where you replace variables with given numbers. It's like swapping out pieces in a game; you take out the 'placeholders' and put in the actual numbers you've been given, which often makes the expression clearer and easier to work with.

For instance, in our exercise, we have the expression 3x(x-2a)^{-x}, where x and a are the variables. The problem tells us to evaluate the expression, providing us with specific values for x and a: x=-2 and a=0.5. By placing these specific values into the expression, we transform the abstract formula into a concrete numerical problem that we can solve step by step.

Here's how it looks when we make those substitutions:
3(-2)((-2)-2(0.5))^{-(-2)}

This process is a fundamental starting point for solving many types of mathematical problems and is crucial in making sure the rest of our work is accurate.

After substituting values, the next step is simplifying expressions. This means performing all the possible operations like addition, subtraction, multiplication, division and simplifying according to the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

Breaking Down Inside the Parentheses

• Deal with operations inside parentheses
• Compute powers or roots
• Multiply or divide from left to right
• Add or subtract from left to right

In our example, the expression 3(-2)((-2)-2(0.5))^{2} simplifies step by step: first, calculations inside the parentheses, then exponents, and finally multiplication. By working through these stages methodically, the expression becomes cleaner and more manageable until it boils down to a single numerical value.

In algebra, exponents and powers refer to the number of times a number, known as the base, is multiplied by itself. The exponent is the small number written slightly above and to the right of the base number, indicating this repetition. Learning how to correctly handle exponents is a key part of simplifying expressions and solving equations.

For example, in the expression (-3)^2, the base is -3, and the exponent is 2, meaning we need to multiply -3 by itself once:
-3 x -3 = 9.

It's also important to remember that a negative base raised to an even exponent results in a positive product, whereas a negative base raised to an odd exponent results in a negative product. This rule is essential to correctly evaluate expressions, as seen in our initial problem where after simplifying, we get (-3)^2, which equals 9, not -9.