Problem 86 - Algebra for College Students

Question

Find each product. Assume that the variables in the exponents represent positive integers. For example, $$ \left(x^{2 n}\right)\left(x^{3 n}\right)=x^{2 n+3 n}=x^{5 n} $$ $$\left(-5 x^{n-1}\right)\left(-6 x^{2 n+4}\right)$$

Step-by-Step Solution

Answer

The product is \(30x^{3n+3}\).

Step 1: Multiply the Constants

First, look at the constant terms in each part of the expression: \((-5) ext{ and } (-6)\)Multiplying these constants gives:\((-5) imes (-6) = 30\).

Step 2: Apply Exponent Rules to the Variables

Next, consider the variable terms, specifically the powers of \(x\). We have:\(x^{n-1} ext{ and } x^{2n+4}\)Since we are multiplying powers of the same base, we can add the exponents:\(x^{(n-1) + (2n+4)}=x^{3n+3}\).

Step 3: Combine Results

Combine the results from Step 1 and Step 2. The product is:\(30x^{3n+3}\).

Key Concepts

Multiplying Exponents

When multiplying exponents, it is important to identify exponents with the same base. In this context, a base is the number or variable that is being raised to a power. When we multiply these powers, we utilize the rule that states:
\[ a^m \times a^n = a^{m+n} \]This means we add the exponents while keeping the base constant.

• For example, if you have \(x^{m}\times x^{n}\), the result will be \(x^{m+n}\).
• This simplifies complex expressions and reduces what might look like an intimidating equation into something more manageable.

In our exercise, we multiply \(x^{n-1}\) and \(x^{2n+4}\). By adding the powers, we simplify this to \(x^{3n + 3}\). This step brings order to algebraic problems, letting us combine separate terms into a single, more elegant one.

Constants Multiplication

Multiplying constants simply involves the multiplication of numbers without any variables. This process translates to the basic arithmetic we are all familiar with.

• Start with the constants from each term of the expression. For instance, in the problem above, the constants \(-5\) and \(-6\) are multiplied together.
• Negative signs are important, as they affect the sign of the product. Multiplying two negative numbers results in a positive product: \((-5) \times (-6) = 30\).

By isolating and multiplying constants separately from variables, we prevent potential confusion and disorder in calculations, allowing us to smoothly proceed to combine results later on.

Algebraic Expressions

Algebraic expressions make up the heart of algebra and are combinations of variables, numbers, and operators. Understanding how to manipulate them is a key part of solving algebraic problems.

• In any algebraic expression, terms are the individual components separated by + or - signs.
• Each term can include numbers (constants), variables (usually represented by letters), and exponents, reflecting the power to which the variable is raised.

In our equation, combining both methods of multiplying exponents and constants results in an expression like \(30x^{3n+3}\). This represents an efficient compilation of separate elements into a unified mathematical statement. Learning to skillfully manage algebraic expressions enhances problem-solving skills and opens up pathways to addressing more complicated algebraic equations in the future.