Problem 79 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The product is \( x^{4n} \).

Step 1: Identify the Base and Exponents

In the expression \( (x^{3n-2})(x^{n+2}) \), the base is \( x \) and the exponents are \( 3n-2 \) and \( n+2 \).

Step 2: Apply the Product of Powers Property

When multiplying terms with the same base, add the exponents. Therefore, add \( 3n-2 \) and \( n+2 \) to find the new exponent: \( 3n-2 + n+2 \).

Step 3: Combine Like Terms in the Exponent

Simplify the expression for the exponent: \( 3n-2 + n+2 = 3n+n-2+2 \). Combine like terms: \( 4n \).

Step 4: Write the Final Simplified Expression

Substitute the simplified exponent back into the expression with the base \( x \). The simplified product is \( x^{4n} \).

Key Concepts

Product of Powers

The product of powers is a fundamental rule in exponents that helps simplify expressions where you multiply terms with the same base. The key principle here is that you add the exponents of the terms. For example, if you have the expression \( x^a \times x^b \), the resulting expression will be \( x^{a+b} \). The base, in this context, remains the same, which is what allows us to conveniently add the exponents.

In our original exercise problem, the expression \( (x^{3n-2})(x^{n+2}) \) showcases the product of powers. The base is \( x \) in both terms. Hence, we add the exponents: \( 3n-2 \) and \( n+2 \), resulting in \( x^{(3n-2)+(n+2)} \).

Following this rule facilitates simplifying complex expressions involving exponents and is a crucial tool in algebra.

Simplifying Expressions

Simplifying expressions in algebra often involves combining like terms and applying rules like the product of powers. This process reduces the expression to its simplest, most straightforward form.

• Identify the operation needed: In our case, the problem required multiplying terms with identical bases, leading us to use the product of powers rule.
• Combine like terms systematically: Add or subtract like terms or coefficients when necessary. For example, the exponents \( 3n-2 \) and \( n+2 \) are combined by adding them: \( 3n+n-2+2 \).
• Perform arithmetic operations: Here, \( 3n+n-2+2 \) simplifies to \( 4n \) as terms \(-2\) and \(+2\) cancel each other out and \( 3n + n \) becomes \( 4n \).

Simplifying expressions allows students to work with easier, manageable forms of algebraic equations, aiding in further calculations and helping to prevent errors.

Mathematical Operations

Mathematical operations are the core of algebraic and arithmetic problem-solving processes. They include addition, subtraction, multiplication, and division. These operations help manipulate and solve equations efficiently. In the given problem \( (x^{3n-2})(x^{n+2}) \):

• We primarily utilize addition to combine exponents. This is part of the approach known as product of powers, where the operation is to add the exponents \( 3n-2 + n+2 \). After simplifying, we get \( 4n \).
• Understanding this process requires basic arithmetic skills, making accurate addition essential in simplifying exponents correctly.

Mastering these math operations is crucial when working with algebraic expressions as they form the foundation for solving more complex problems in mathematics.