Problem 105 - Elementary and Intermediate Algebra

Question

Translate each phrase to mathematical symbols. Let \(x\) represent the unknown number. a. The sum of a number and 7 squared b. The sum of a number and 7 , squared

Step-by-Step Solution

Answer

a) \(x + 7^2\); b) \((x + 7)^2\)

Step 1: Analyze the Phrase

The first phrase is 'The sum of a number and 7 squared'. We must carefully distinguish between summing first and squaring the result, versus squaring the 7 first. Here, 7 is squared before adding to the unknown number.

Step 2: Translate to Mathematical Symbols

Add the squared number 7 to the unknown number, which is expressed as \(x + 7^2\). This denotes the addition of the squared 7 to \(x\).

Step 3: Analyze the Second Phrase

The second phrase is 'The sum of a number and 7, squared'. This indicates that the sum of the unknown number and 7 is first calculated, and then the entire result is squared.

Step 4: Translate to Mathematical Symbols for Second Phrase

Write the sum of the number and 7 first, then square the entire expression. This is expressed as \((x + 7)^2\), where the parentheses imply squaring the result of the addition.

Key Concepts

Mathematical Symbols

Mathematical symbols are essential tools that help us express complex ideas and calculations in a concise and universal manner. They act as a language, allowing us to translate verbal statements into precise mathematical expressions. For instance,

• The symbol "+" stands for addition.
• The symbol "^2" indicates squaring a number, which means multiplying the number by itself.
• Brackets like "()" are used to denote priority in operations.

In our example, when translating phrases into mathematical symbols, we use "x" to represent an unknown number. This is common practice in algebra, where letters are often used to make it easier to handle and solve problems involving unknown values.
Understanding mathematical symbols helps to avoid misunderstandings, as it often determines the correct sequence of operations in an expression.

Order of Operations

Order of operations is a fundamental concept in mathematics that dictates the sequence in which different operations should be performed to accurately solve expressions. **PEMDAS** is a common acronym used to remember this order:

• **P**: Parentheses - Solve expressions within brackets first.
• **E**: Exponents - Next, compute powers (e.g., squared numbers).
• **MD**: Multiplication and Division - Perform these from left to right.
• **AS**: Addition and Subtraction - Lastly, execute these from left to right.

Consider two phrases: 'The sum of a number and 7 squared' translates to \(x + 7^2\). Here, we square 7 before adding it to \(x\). Alternatively, 'The sum of a number and 7, squared' gives us \((x + 7)^2\), where the addition is done first due to the parentheses, followed by squaring the result. This order is critical to achieving the correct solution.

Algebraic Expressions

Algebraic expressions are mathematical phrases that include numbers, variables, and operational symbols. These expressions can represent real-world scenarios, enabling us to solve problems by performing various operations. Let's look at the example phrases given:

• The phrase 'The sum of a number and 7 squared' leads to the algebraic expression \(x + 7^2\). This expression indicates that the number 7 is squared before being added to an unknown number \(x\).
• The phrase 'The sum of a number and 7, squared' translates to \((x + 7)^2\). Here, the sum \(x + 7\) is calculated first, and then the entire result is squared.

Algebraic expressions often require understanding the relationships between numbers and operations, using variables to simplify and solve equations. Practicing translation from verbal phrases to algebraic expressions is crucial since it forms the foundation of more complex problem-solving in algebra.