Problem 99 - Elementary Algebra

Question

For Problems \(93-106\), evaluate each algebraic expression for the given values of the variables. $$ -x^{2} \quad \text { for } x=-8 $$

Step-by-Step Solution

Answer

The expression evaluates to -64.

Step 1: Substitute the Value of x

Substitute the given value of the variable into the algebraic expression. We have the expression \(-x^2\) where \(x = -8\). First, substitute \(-8\) in place of \(x\) to get \(-(-8)^2\).

Step 2: Simplify the Expression Inside the Parentheses

Evaluate the term inside the parentheses first, so calculate \((-8)^2\). Raising \(-8\) to the power of 2 means multiplying \(-8\) by itself: \((-8) \times (-8) = 64\).

Step 3: Apply the Negative Sign

Apply the negative sign outside the bracket to the result obtained from Step 2. So, we have \(-((-)8)^2 = -64\).

Step 4: Conclusion

The expression \(-x^2\) evaluates to \(-64\) when \(x = -8\).

Key Concepts

Substitute Value

Substituting a value into an algebraic expression is like finding a missing puzzle piece. You replace the variable in the expression with the number provided.

Let's break this down using our example, where we have the expression \(-x^2\) and our variable \(x\) is given as \(-8\). Therefore, to substitute is to simply replace \(x\) with \(-8\). It's like unwrapping a gift to see what's inside: you have \(-(-8)^2\).

• Identify the expression, which is \(-x^2\).
• Notice that \(x = -8\), so put \(-8\) in place of \(x\).

Once you substitute \(-8\) for \(x\), it sets the stage for simplifying the equation. Carrying out this step correctly ensures the rest of your computation flows smoothly.

Simplify Expression

Simplifying an expression is all about making it easier to understand and work with. It often involves following the correct order of operations.

For our expression, after substitution, we are left with \(-(-8)^2\). Now, the first task is to handle the term inside the parentheses:

• First, evaluate \((-8)^2\).
• This means multiplying \(-8\) by itself: \((-8) \times (-8)\).
• The result is \(64\), because multiplying two negative numbers results in a positive number.

Simplifying is crucial because it reduces room for mistakes and clarifies your results, turning our complex expression into something that's easy to work with. This step is like clearing the fog so you can see the road ahead clearly.

Negative Sign Application

Applying the negative sign effectively is important especially when dealing with expressions that involve powers.

After simplifying the expression inside the parentheses, we were left with \(64\). Our full expression becomes \(-64\), as there is a negative sign in front of the entire expression.

• Take the simplified result, which is \(64\).
• Remember the negative sign outside the bracket from the original expression \(-x^2\).
• Apply the negative sign to the \(64\), resulting in \(-64\).

This step reintroduces the negative sign at the very end, making sure that we stay true to the original problem. It's like adding the sour twist that makes a lemonade perfect, ensuring that the final answer makes sense and corresponds with the expression you began with.