Problem 41 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The simplified expression is 16.

Step 1: Simplify Exponents

Start by calculating each of the exponents in the expression:- Calculate \( 2^3 = 8 \).- Calculate \( (-1)^3 = -1 \).- Calculate \((-2)^2 = 4 \) and then \( 3 imes (-1) = -3 \), so \( 3(-1)^3(-2)^2 = -3 imes 4 = -12 \).- Calculate \( (2)^2 = 4 \) and then \( -5 imes (-1) = 5 \), so \( -5(-1)(2)^2 = 5 imes 4 = 20 \).

Step 2: Substitute Results Back

Substitute the calculated values back into the expression:\[ 8 - 12 + 20 \]

Step 3: Perform Addition and Subtraction

Finally, simplify the expression by performing the addition and subtraction:- Start with \( 8 - 12 = -4 \).- Then add \( -4 + 20 = 16 \).

Key Concepts

Simplify Exponents

Exponents are a shorthand way to express repeated multiplication, and simplifying them is a key step in solving numerical expressions. When you see a number raised to an exponent, it means you're multiplying that number by itself a certain number of times. For example, in the expression \(2^3\), the base is 2, and the exponent is 3, meaning we multiply 2 by itself three times:

• \(2 \times 2 \times 2 = 8\)

When handling negative bases, like \((-1)^3\), the rules are slightly different. Negative numbers raised to odd exponents remain negative. Hence, \((-1)^3 = -1\). In contrast, negative numbers raised to even exponents become positive, as shown in \((-2)^2 = 4\), because multiplying two negatives yields a positive.Understanding these rules helps when simplifying expressions involving exponents. It is the groundwork upon which you will build the rest of the solution, so make sure you fully grasp it before moving on.

Addition and Subtraction

Once the exponents in an expression are simplified, the next step is to tackle addition and subtraction. This is where you combine the results to form a simpler expression. It helps to follow the left-to-right rule and deal with each operation as you encounter it in the expression.Take this example from the step-by-step solution:

• First, simplify the term \(8 - 12 = -4\).
• Then, proceed with \(-4 + 20 = 16\).

Breaking the operation down this way ensures clarity and prevents mistakes. When dealing with several operations at once, organizing your work in steps helps to maintain control and accuracy. This practice also makes it easier to spot where things might have gone wrong if you end up with an unexpected result.Addition and subtraction are fundamental operations, but proficiency comes with practice and careful manipulation of numbers and signs to achieve the correct outcome.

Step by Step Solution

A methodical, step-by-step approach to solving mathematical problems ensures that you do not overlook any essential calculation or rule. Following a structured process leads to a correct and easy understanding of the solution. Let's revisit the given problem:Firstly, we broke down the expression by individually simplifying its components, such as \(2^3\), \((-1)^3\), and \((-2)^2\). This isolated approach allows you to focus on one piece at a time, reducing the complexity of the problem as a whole.Next, substituting these simplified components back into the expression like so:

• Calculated expression: \(8 - 12 + 20\).

Finally, the process culminated in solving addition and subtraction in sequence, through careful arithmetic steps: first \(8 - 12 = -4\), then following this with \(-4 + 20 = 16\). This step-by-step execution confirms the correctness of your answer, consolidating the applied logic from start to finish.Adopting such systematic methods helps tackle even the most complicated expressions with confidence while instilling good habits in problem-solving.