Problem 47 - Intermediate Algebra

Question

Simplify each of the numerical expressions. $$ 2(-1)^{3}-3(-1)^{2}+4(-1)-5 $$

Step-by-Step Solution

Answer

The simplified expression is \(-14\).

Step 1: Calculate \((-1)^3\)

First, compute \((-1)^3\). Raising \(-1\) to an odd power results in \(-1\), so \((-1)^3 = -1\).

Step 2: Calculate \((-1)^2\)

Next, compute \((-1)^2\). Raising \(-1\) to an even power results in \(1\), so \((-1)^2 = 1\).

Step 3: Multiply Each Term

Now, substitute the results from Steps 1 and 2 back into the expression:\[2(-1) - 3(1) + 4(-1) - 5\]This becomes:\[-2 - 3 - 4 - 5\].

Step 4: Simplify the Expression

Now, add up the terms:1. First, add \(-2\) and \(-3\), which equals \(-5\).2. Then, add \(-5\) and \(-4\), which equals \(-9\).3. Finally, add \(-9\) and \(-5\), which equals \(-14\).Therefore, the expression simplifies to \(-14\).

Key Concepts

Understanding Exponents

Exponents are a way to express repeated multiplication of a number by itself. They are written as a small number, called the exponent, placed at the top right corner of another number, known as the base. For example, in \((-1)^3\), the base is -1, and the exponent is 3. This expression tells you to multiply -1 by itself three times:

• \((-1) \times (-1) \times (-1) = -1\)

This is because the product of a negative number raised to an odd power remains negative. Conversely, when a negative number is raised to an even power, such as \((-1)^2\), the result is 1, because the negative signs cancel each other out:

• \((-1) \times (-1) = 1\)

Exponents play a crucial role in simplifying expressions, helping you solve them step by step.

Simplification of Expressions

Simplifying expressions involves reducing them to their most basic form. This means replacing complex operations or multiple terms with simpler ones. Let's look at the expression from the exercise:

• First, substitute the values of exponents: \[2(-1) - 3 \times 1 + 4(-1) - 5\]
• Next, perform the multiplication in each term: \[-2 - 3 + (-4) - 5\]
• Now, combine the results: \[-2 - 3 - 4 - 5\]

Simplifying involves doing these operations in the right order, known as "order of operations" (PEMDAS/BODMAS). Always solve exponent terms first, followed by multiplication, and finally addition or subtraction, effectively reducing the expression step by step to arrive at a final answer.

Working with Negative Numbers

Negative numbers are numbers that are less than zero, represented with a minus sign (-). When simplifying expressions involving negative numbers, it's important to pay attention to their interactions. For example:

• When adding two negative numbers, the result is more negative: \(-2 + (-3) = -5\)
• When subtracting a larger number from a smaller number, you end up with a negative result: \(-3 - 5 = -8\)

Understanding these interactions is crucial for correctly simplifying expressions. Each operation affects the overall value differently, so keeping track of signs and numbers aids in correctly computing the final result. In our simplified expression, continual subtraction of negative numbers results in a larger negative value of \(-14\). Negative numbers, when combined, influence the outcome significantly, emphasizing the importance of precise calculation.