Problem 30 - Elementary Algebra

Question

Rewrite the problem in a simpler form. $$ -(-8) $$

Step-by-Step Solution

Answer

Answer: 8

Step 1: Understand the given expression

We are given the expression: ``` -(-8) ``` There is a double negative involved here, which means we have a negative sign and inside the parenthesis, we have another negative sign in front of the number 8.

Step 2: Apply the double negative rule

When we have a double negative, that means multiplying two negative numbers. In general, when we multiply two negative numbers, we get a positive number. So, we can rewrite our expression as: ``` -(-8) = 1 * 8 ```

Step 3: Simplify the expression

Now that we have rewritten our expression, let's simplify it by performing the multiplication: ``` 1 * 8 = 8 ``` The simplified expression is 8. So, -(-8) simplifies to 8.

Key Concepts

Negative Numbers

Negative numbers are those less than zero, often represented by a minus sign (-) before the number. When working with negative numbers, one useful rule is that a negative sign inverts the value of a number. For instance, while 8 is positive, -8 is its negative equivalent. These numbers exist on the left side of a number line and are usually related to concepts such as debt or temperature below zero.

When two negative numbers interact, particularly through multiplication, some specific rules apply. A double negative, such as the expression \(-(-8)\), cancels out the negativity. Think of it as removing an additional layer of "negative" to find the true value beneath. This basic principle is used frequently in simplifying expressions in algebra.

Simplifying Expressions

Simplifying expressions in algebra means making an expression easier to understand and work with. An expression like \(-(-8)\) initially looks complicated due to the double negative. The goal of simplifying is to break down complex expressions using mathematical properties and rules.

In our example, converting \(-(-8)\) into a simpler form means first understanding that the two negatives cancel each other out. By acknowledging this, we can directly transform the expression to its positive equivalent, which is 8. This process can also extend to combining like terms or reducing fractions but here it focuses on merely correcting signs and making the number appear positive.

Multiplication Rules

Multiplication involving negative numbers follows specific straightforward rules. When you multiply two negative numbers, the result is positive. It might seem counterintuitive, but you can visualize it as reversing two times. One negative reverses the direction, and applying another negative reverses it yet again, bringing it back to the original positive direction.

Consequently, in expressions, these rules mean that \(-1 \times -8\) equals positive 8, which simplifies further in solving problems. The multiplication rules form a core part of algebra and arithmetic, helping students accurately solve equations and expressions with multiple layers of numerical data. These principles underpin various calculations you'll encounter frequently in algebraic contexts.