Problem 33 - Fundamentals of Mathematics

Question

For problems \(17-46\), find the value of each expression. $$ -k^{2}-k-1, \text { if } k=-1 $$

Step-by-Step Solution

Answer

The value of the expression is -1.

Step 1: Substitute the Value of k

Given the expression \[-k^{2}-k-1\] and the value of \(k = -1\), substitute \(-1\) for \(k\) in the expression. This gives:\[-(-1)^{2} - (-1) - 1\]

Step 2: Simplify k Squared

Calculate \((-1)^2\):\[-(-1)^2 = -1\] This is because squaring any negative number gives a positive result.

Step 3: Simplify k Term

Simplify the term \(- (-1)\):\[- (-1) = +1\] The negative of a negative number is positive.

Step 4: Combine the Expression

Combine the simplified parts of the expression:\[-1 + 1 - 1\] Follow the order of operations to combine these terms step by step.

Step 5: Final Calculation

Perform the calculation:\[-1 + 1 = 0\] Now subtract 1:\[0 - 1 = -1\]

Key Concepts

Substitution Method

In algebra, the substitution method is a powerful tool to simplify expressions and equations. It involves replacing variables with their given values and then simplifying the expression. In the original problem, the variable is \(k\) and its value is \(-1\). The expression is \(-k^2-k-1\). To apply the substitution method:

• First, substitute \(-1\) for every occurrence of \(k\) in the expression. This leads to replacing \(k\) with \(-1\), resulting in \(-(-1)^2 - (-1) - 1\).
• This step is crucial because it allows us to transform an algebraic expression into a numeric calculation that can be simplified further.

This method simplifies the process of solving problems involving algebraic expressions by transforming them into simpler arithmetic forms.

Order of Operations

The order of operations is essential when simplifying complex expressions. It dictates the sequence in which operations should be performed to ensure accuracy. When dealing with the expression \(-(-1)^2 - (-1) - 1\), we follow these steps:

• First, calculate the exponent \((-1)^2\), which equals \(1\).
• Next, handle the negation before the term, which applies throughout the entire calculation. So, we calculate \(-(-1)^2\).
• Subsequently, perform addition and subtraction from left to right. In this case, it involves resolving \(-1 + 1 - 1\).

Using the order of operations helps avoid mistakes that can arise from simply going from left to right without considering hierarchy among arithmetic operations. Following this systematic approach ensures we get the correct result every time.

Negative Numbers

Negative numbers often present a challenge, especially in algebraic expressions. Understanding the rules of negative numbers is crucial when dealing with expressions like \(-(-1)^2 - (-1) - 1\). Consider these points:

• Squaring any negative number results in a positive number, as in \((-1)^2 = 1\).
• Multiplying or dividing two negative numbers results in a positive number.
• The negative of a negative number is positive, meaning \(-(-1) = +1\).

Complete understanding of these principles ensures clarity and precision when simplifying expressions involving negative numbers. This knowledge not only helps in the given exercise but also in diverse mathematical problems.

Exponents

Exponents are a critical component of many algebraic expressions. They indicate how many times a number, known as the base, is multiplied by itself. In the expression \(-(-1)^2 - (-1) - 1\), exponents play a key role:

• Understand that exponentiation is performed before multiplication or division according to the order of operations.
• In the example, the base \(-1\) is squared to produce \((-1)\times(-1)=1\).
• Exponents must be computed at the outset to influence the subsequent operations correctly.

Recognizing and correctly applying exponents allows for the simplification of potentially complicated expressions into manageable calculations. This vital skill is fundamental for moving forward in algebra and beyond.