Problem 91 - Introductory Algebra for College Students

Question

In Exercises \(89-92,\) 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

36

Step 1: Translate Words into Numerical Expression

Read each part of the sentence and translate it to math. 'Subtract 10 from 7' becomes \(7 - 10\), 'Multiply this difference by 2' becomes \((7 - 10) \times 2\), 'Square this product' becomes \(((7 - 10) \times 2)^2\). So, the full numerical expression is \(((7 - 10) \times 2)^2\)

Step 2: Order of Operations and Simplification

Simplify the expression by following the order of operations. First, calculate subtraction inside the parentheses: \((-3 \times 2)^2\). Next, calculate multiplication: \((-6)^2\). Finally, calculate the square: \(36\).

Step 3: Write the Final Answer

The simplified numerical expression is \(36\). Write this as the final answer.

Key Concepts

Numerical Expression

Understanding how to frame a numerical expression is a crucial first step in solving many algebraic problems. A numerical expression is a combination of numbers and operations that, when simplified, can produce a single number as an answer.

For example, let's consider the exercise that asks to 'Subtract 10 from 7, multiply this difference by 2, and square this product.' This sentence is translated into the numerical expression \(((7 - 10) \times 2)^2\). Here, we have subtraction \( (7 - 10) \), multiplication \( (\cdot \times 2) \), and exponentiation \( (\cdot)^2 \). It's important to pay attention to the order in which these operations are laid out, as this will affect the final result.

The use of parentheses in the expression is crucial. It clearly indicates which operation should be performed first, thus avoiding any misinterpretation when it comes to the order of operations.

Simplifying Expressions

When we simplify an expression, it means we are performing all the operations in the correct order to find the simplest form or most reduced version of that expression. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), guides us in this process.

Following the order, we first solve anything inside parentheses, then compute exponents, followed by any multiplication or division from left to right, and lastly tackle any addition or subtraction also from left to right. The PEMDAS rule simplifies the given expression <\text=((7 - 10) \times 2)^2\text==> to produce a final value of 36. In our example, we've followed these steps carefully:

• Handle the subtraction inside the parentheses: \((7 - 10) = -3\)
• Follow with the multiplication: \( (-3 \times 2) = -6 \)
• Lastly, apply the square: \( (-6)^2 = 36 \)

By understanding and applying this hierarchy of operations, expressions are broken down systematically and solved accurately.

Algebraic Problem-Solving

In the realm of algebraic problem-solving, translating words into numerical expressions and simplifying them are foundational skills. Problem-solving involves breaking down a problem into steps that are easier to understand and execute.

By starting with a clear phrase that describes the problem, such as in our exercise, and translating it into mathematical terms, we turn the abstract into the concrete. The given problem's phrase 'Subtract 10 from 7, multiply this difference by 2, and square this product' is a sequential guide for the operations needed.

Problem-solving, in this context, means following those steps accurately:

• Translate the sentence into a numerical expression.
• Use the order of operations to simplify the expression step by step.
• Reach the final numerical answer, clear in its simplicity.

Practicing these skills in a variety of problems will strengthen your ability to approach and solve more complex algebraic equations. Every problem is an opportunity to master these steps, one operation at a time.