Problem 6 - Prealgebra

Question

Write each expression using exponents. $$(a+1)(a+1)$$

Step-by-Step Solution

Answer

The expression \((a+1)(a+1)\) can be written as \((a+1)^2\).

Step 1: Identify the Expression Form

The given expression is \((a+1)(a+1)\).This expression represents the product of the same base, \((a+1)\), multiplied by itself.

Step 2: Recognize the Exponents Pattern

When multiplying the same base together, the exponents of the base can be represented by the number of times the base appears in the product. Here, \((a+1)\) appears twice.

Step 3: Convert to Exponential Form

Since \((a+1)\) multiplies itself, we can rewrite the expression in exponential form as \((a+1)^2\). This uses the rule that multiplying a number or expression by itself is the same as raising it to the power of 2.

Key Concepts

Understanding Algebra in Expressions

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It allows us to express and solve a wide variety of real-world problems. In algebra, we frequently use expressions to represent mathematical relationships.

An expression in algebra is a combination of numbers, variables, and operation symbols. In the original exercise, \((a+1)(a+1)\) is an algebraic expression where:

• \(a\) is a variable that may represent any number.
• \(+1\) is a constant added to the variable.
• \((a+1)\) is an entire expression being multiplied by itself, showing multiplication of an identical structure.

Using algebra, we can transform these structures by applying rules and properties, such as the use of exponents, to simplify and better understand them.

What is an Exponential Expression?

An exponential expression involves a base and an exponent. The exponent tells how many times the base is used as a factor in a multiplication. This is a compact way to express repeated multiplication, which is common in algebra.

In our example, \((a+1)(a+1)\), we recognize that the expression repeats the base \((a+1)\) twice. This repetition allows us to combine them using an exponent. We can write an exponential expression as \((a+1)^2\), where:

• \((a+1)\) is the base.
• \(2\) is the exponent, indicating the base is multiplied by itself.

This transformation from multiplied terms to an exponential form helps to manipulate the expression more efficiently, a key skill developed in algebra.

The Role of Mathematical Notation

Mathematical notation is a system of symbols and signs used to represent numbers, functions, relations, and operations concisely. It is essential in conveying complex mathematical ideas swiftly and clearly.

Mathematical notation is particularly useful when dealing with exponential expressions. The notation \((a+1)^2\) uses:

• The parentheses to signify that the operation inside \((a+1)\) is considered a single entity or base.
• The superscript \(^2\) indicating the number of times the base is multiplied by itself.

This compact form allows mathematicians to work with large or complex calculations effortlessly, making problem-solving more manageable. Understanding and using this notation is a fundamental skill in algebra and mathematics.