Problem 87 - Elementary Algebra

Question

Simplify. $$ -(-(1)) $$

Step-by-Step Solution

Answer

The simplified expression is 1.

Step 1: Understanding the Inner Expression

Identify the expression within the innermost set of parentheses. Inside the parentheses, we have a simple number: 1.

Step 2: Negate the Inner Expression

Apply the negation (negative sign) to the inner expression: - (1) = -1. This means that negating 1 gives -1.

Step 3: Apply the Outer Negation

Now apply the negation to the result from Step 2. So we have: -(-1). Negating -1 results in the positive version of the number; therefore, -(-1) = 1.

Key Concepts

Negation

Negation in algebra, often represented as the negative sign, can change the sign of a number or an expression. When you see a negative sign in front of a number, it implies the opposite of that number's value.
For example, \(-1\) is the negation of \(1\). Applying negation twice, as in \(-(-1)\), reverses the sign back to its original state. This concept is crucial when dealing with expressions within parentheses, where each negation flips the sign.
In practical terms:

• One negation turns a positive into a negative, like \(5\) to \(-5\).
• An extra negation flips it again, converting \(-5\) back to \(5\).

Remember that negation doesn’t change the absolute value; it only affects the sign.

Parentheses Evaluation

Parentheses in algebraic expressions indicate which parts of an expression should be evaluated first. They help structure equations and set a specific order of operations. Evaluating the expression within these parentheses is a vital step before handling other parts of the equation.
For example, in the expression \(-(-(1))\), the parentheses dictate that we should first look at the number \(1\) inside the innermost set. This hierarchy ensures that calculations follow a logical progression.
If there are multiple layers of parentheses, work your way from the inside out. Always handle operations in the following sequence:

• Solve expressions within parentheses or brackets first.
• Consider any exponents or roots after that.
• Then proceed with multiplication or division.
• Finally, perform addition or subtraction.

This systematic approach simplifies complex algebraic expressions and avoids errors.

Simplification Steps

Simplification is the process of reducing an algebraic expression to its simplest form without changing its value. This technique makes expressions easier to work with and understand.
In the given example \(-(-(1))\), the simplification steps are straightforward but essential:

• Identify the expression inside the innermost parentheses, which is \(1\).
• Apply negation to this value, leading to \(-1\).
• Apply the outer negation, making the expression \(-(-1)\), which simplifies to \(1\).

Breaking down each step ensures accuracy and clarity, highlighting how each part of the expression contributes to the final simplified form.