Problem 91 - Introductory and Intermediate Algebra for College Students 4th

Question

Express each sentence as a single numerical expression. Then use the order of operations to simplify the expression Subtract 10 from \(7 .\) Multiply this difference by \(2 .\) Square this product.

Step-by-Step Solution

Answer

The simplified numerical expression is 36.

Step 1: Translate to Arithmetic Expression

First, translate the given sentence to an arithmetic expression. Here, 'Subtract 10 from 7' translates to \(7-10\). 'Multiply this difference by 2' translates to \((7-10) \times 2\). 'Square this product' translates to \(((7-10) \times 2)^2\). So, the entire sentence translates to \(((7-10) \times 2)^2\)

Step 2: Simplify Inside Parentheses

Apply the operation inside the parentheses. We start with \(7 - 10\) which results in \(-3\). Replace \(7 - 10\) in the expression with \(-3\). The expression now becomes \((-3 \times 2)^2\).

Step 3: Multiply

Next, multiply \(-3\) by \(2\). The result is \(-6\). Substitute this result into the expression, resulting in \((-6)^2\)

Step 4: Square the Result

Lastly, raise \(-6\) to the power of \(2\) to get \(36\). Therefore, the simplified expression is \(36\)

Key Concepts

Numerical Expression

A numerical expression is a combination of numbers and operators (such as addition, subtraction, multiplication, and division) that represents a particular value. In mathematics, a numerical expression could be as simple as a single number or a more complex combination like \(3 + 4 \times 2\), which requires calculations to find the value it represents.

When dealing with numerical expressions, it is essential to follow the conventional order of operations to arrive at the correct result. This order is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

The exercise given presents a sentence-based problem that needs to be translated into a numerical expression for further simplification. Initially, the problem involves performing operations sequentially: subtracting, multiplying, and then squaring. Understanding how to correctly convert sentences into a numerical expression helps in accurately solving problems.

Arithmetic Expression

An arithmetic expression is a mathematical phrase that contains numbers, operators, and sometimes parentheses, combining them to produce a value.

In the original exercise, the arithmetic expression is \( ((7-10) \times 2)^2 \), which represents a series of operations that need to be performed in a specific order to solve the problem. Translating words into an arithmetic expression involves identifying the mathematical operations described and then creating an equation that accurately mirrors those steps.

Let's break down the expression:

• First, subtract 10 from 7: \(7 - 10\).
• Then, multiply the result by 2: \((7-10) \times 2\).
• Lastly, square the result: \(((7-10) \times 2)^2\).

This structured approach to forming expressions from words makes it easier to use mathematical operations correctly and reach the solution accurately.

Simplification

Simplification is the process of performing operations in a step-by-step manner to reach a concise and easiest-to-read form of an answer. In mathematics, simplification reduces complex expressions to simpler forms or a single number.

In the exercise provided, simplification involves multiple steps: reducing the operations inside the parentheses, multiplying the results, and finally squaring the product. Let's see how this occurs:

• Start by solving the operation within the parentheses, which is \(7 - 10\), resulting in \(-3\).
• Next, multiply \(-3\) by 2, yielding \(-6\).
• Finally, square \(-6\) to obtain the result, which is 36.

The aim of simplification is to make complex calculations more manageable and the final result easier to understand. Each operation follows a structured order, ensuring that the mathematical processes align with the expected outcome.

Parentheses in Mathematics

Parentheses are a crucial part of mathematics as they help denote which operations should be performed first in an expression. This is especially important when dealing with complex expressions that involve multiple operations.

In our example, the role of parentheses can be seen in the expression \( ((7-10) \times 2)^2 \). The parentheses around \(7 - 10\) dictate that we must subtract first before moving on to multiplication. Later, after multiplication, squaring the result comes next. This ensures that the operations enclosed within parentheses are prioritized and solved ahead of others, following the rules of the order of operations.

• Using parentheses clarifies which calculations are grouped together.
• Ensures accuracy in executing the correct sequence of operations.
• Helps in avoiding miscalculations by following established mathematical norms.

Mastering the use of parentheses in mathematical expressions is essential for solving problems correctly and efficiently.