Problem 78 - Algebra for College Students

Question

Find each product. Assume that the variables in the exponents represent positive integers. For example, $$ \left(x^{2 n}\right)\left(x^{3 n}\right)=x^{2 n+3 n}=x^{5 n} $$ $$\left(a^{5 n-1}\right)\left(a^{5 n+1}\right)$$

Step-by-Step Solution

Answer

The product is \(a^{10n}\).

Step 1: Identify the Base and Exponents

First, recognize that both terms share the same base, which is \(a\). The exponents are \(5n-1\) for the first term and \(5n+1\) for the second term.

Step 2: Apply the Product of Powers Property

According to the Product of Powers Property, when multiplying two expressions with the same base, you add the exponents. Therefore, the expression becomes: \(a^{(5n-1) + (5n+1)}\).

Step 3: Simplify the Exponent

Now, simplify the exponent by combining like terms: \((5n-1) + (5n+1) = 5n - 1 + 5n + 1 = 10n\).

Step 4: Write the Simplified Expression

Since the simplified exponent is \(10n\), the expression can now be written as: \(a^{10n}\).

Key Concepts

Understanding Exponents

In algebra, exponents signify how many times a number, known as the base, is multiplied by itself. For example, in the expression \(a^3\), \(a\) is the base, and 3 is the exponent, which means \(a \times a \times a\). Exponents are a shorthand way to express repeated multiplication and have several properties that simplify arithmetic operations.

• They make it easier to work with very large or very small numbers.
• They help simplify multiplication when dealing with variables in algebra.

One of the significant properties of exponents is the Product of Powers Property, which states that when you multiply two expressions with the same base, you simply add the exponents together. This is crucial for efficiently solving problems involving powers.

Simplifying Expressions with Exponents

Expressions with exponents can be simplified using several exponent rules, making calculations easier. In the given exercise, you're tasked with simplifying expressions that involve multiplying terms with the same base, like \((a^{5n-1})(a^{5n+1})\).To simplify such expressions, use the Product of Powers Property. This property simplifies the process by allowing you to add the exponents:\[a^{(5n-1) + (5n+1)}\] When simplifying, don't forget to combine like terms. In this case, the exponents \(5n-1\) and \(5n+1\) add up to \(10n\). Thus, the expression simplifies to a more manageable form:\[a^{10n}\]By understanding these steps, you can simplify other expressions with similar structures systematically. It saves time and reduces errors.

Exploring Algebraic Multiplication

Algebraic multiplication involves operations between numbers and variables, often using exponents to express repeated multiplication. The key to mastering algebraic multiplication lies in understanding how to manipulate these expressions using specific rules.Imagine multiplying two numbers, where each number is a power of the same base. Instead of lengthy calculations, apply multiplication properties, which involve simple arithmetic on the exponents. Steps involved include:

• Identifying the terms' base and exponent clearly.
• Applying rules like the Product of Powers seamlessly.
• Simplifying the expression by arithmetic operations on the exponents.

In our example, the expression \((a^{5n-1})(a^{5n+1})\) can further be reduced because it adheres to the Product of Powers rule. The multiplication becomes a straightforward addition problem for the exponents, leading to \[a^{10n}\]. Mastering these rules and applying them effectively is essential for excelling in algebra.