Problem 41 - Algebra for College Students

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: Calculate the Powers

First, address any exponents in the expression. Calculate each power separately: \[2^{3} = 8, \quad (-1)^{3} = -1, \quad (-2)^{2} = 4, \quad (2)^{2} = 4.\]

Step 2: Substitute the Values into the Original Expression

Replace the powers in the original expression with the calculated values:\[8 + 3(-1)(4) - 5(-1)(4).\]

Step 3: Perform the Multiplication

Multiply the numbers in each term: For the second term: \[3 \cdot (-1) \cdot 4 = -12.\] For the third term: \[5 \cdot (-1) \cdot 4 = -20.\]

Step 4: Substitute Back and Simplify

Insert these calculated products back into the expression to simplify:\[8 + (-12) - (-20).\]Simplify by performing the addition and subtraction in order:\[8 - 12 + 20.\]

Step 5: Final Simplification

Perform the arithmetic operations from left to right:\[8 - 12 = -4.\] \[-4 + 20 = 16.\] Thus, the simplified expression is \[16.\]

Key Concepts

Numerical Expressions

When we deal with numerical expressions, we are simply working with combinations of numbers and operations like addition, subtraction, multiplication, and division, as well as groupings such as parentheses and exponents. Understanding numerical expressions is crucial because they form the foundation of algebra and higher math topics.

In this particular exercise, you encountered a numerical expression that includes not just basic operations but also exponents:

• Each part of the expression represents a different number derived either by direct observation or calculation.
• The expression provided in the problem represents a combination of operations and numbers, organized in a structured way.

Numerical expressions can be complex, but by breaking them down into simpler parts, such as individual power calculations and basic arithmetic operations, it becomes more manageable to simplify an entire expression. Remember, numerical expressions are simply detailed instructions read from left to right to perform specific calculations.

Order of Operations

The term 'order of operations' refers to a set of rules that dictate the order in which mathematical operations should be carried out to ensure consistent and correct results. These rules are sometimes remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

In our exercise, we apply these rules as follows:

• **First**, we address all the exponents in the expression, as the order of operations dictates handling powers before any multiplications or additions.
• **Next**, you deal with multiplication and division, which might often appear after you've handled any exponents, from left to right.
• **Lastly**, you perform addition and subtraction in the order they appear.

Thus, by following these steps, we ensure that no matter who simplifies the expression, the outcome will always be the same.

Simplification Process

Simplifying expressions involves condensing a complex expression into its simplest form while maintaining the same value. When simplifying, especially in algebra and arithmetic, each step is crucial for arriving at the correct result.

In our exercise:

• **Initially**, we calculated each exponent. This gave values that we can directly substitute back into the expression.
• **Then**, multiplication was performed, simplifying the expression further by reducing each part marked by parentheses.
• **Finally**, all arithmetic operations, like addition and subtraction, were methodically carried out to simplify the original expression into a single integer.

Just like solving a puzzle, the simplification process involves orderly steps that condense information into a clearer, often single-term, answer. This approach ensures precision and clarity in mathematical problem-solving, making it easier not only for us to reach our solution but also to verify it against established solutions.