Problem 80 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The product is \(x^{5n+2}\).

Step 1: Understanding the Product Rule

When you multiply expressions with the same base, the exponents are added. This is known as the Product Rule for exponents. Mathematically, this is expressed as \(a^m \cdot a^n = a^{m+n}\).

Step 2: Identify the Base and Exponents

In the expression \((x^{n-1})(x^{4n+3})\), the base is \(x\). The exponents are \(n-1\) and \(4n+3\).

Step 3: Apply the Product Rule

Add the exponents as per the Product Rule: \(n-1 + 4n+3\). This simplifies to \((n-1) + (4n+3) = 5n + 2\).

Step 4: Write the Final Product

Using the result from Step 3, write the final expression: \(x^{5n+2}\).

Key Concepts

Product Rule

The Product Rule is a fundamental rule in algebra that helps us manipulate expressions involving exponents. It applies when you multiply two expressions with the same base, such as - \(a^m \cdot a^n\).The Product Rule states that when these expressions are multiplied:

• You keep the common base.
• Add together the exponents.

This is mathematically expressed as \(a^{m+n}\). For example, if you have \(x^2 \cdot x^3\), applying the Product Rule gives you \(x^{2+3}\), which simplifies to \(x^5\).This rule simplifies the process of working with powers and is especially useful in solving complex problems that involve multiplying terms with the same base.

Exponents

Exponents serve as a shorthand way to express repeated multiplication of a number by itself. In the expression \(x^n\), the base is \(x\) and the exponent \(n\) indicates that \(x\) is multiplied by itself \(n\) times. Exponents help simplify expressions and make calculations easier. Common properties of exponents include:

• Zero Exponent: \(a^0 = 1\) for any non-zero \(a\).
• Negative Exponent: \(a^{-n} = \frac{1}{a^n}\).
• Power of a Power: \(a^{m^n} = a^{m \cdot n}\).

Understanding how exponents work is crucial for applying other rules, such as the Product Rule. This allows you to handle more complex expressions with confidence.

Algebraic Expressions

Algebraic expressions combine numbers, variables, and mathematical operators to represent values. They consist of terms, which are parts of an expression separated by addition or subtraction. Each term can include:- A constant (like 5).- A variable (like \(x\)).- A product of a constant and variables raised to powers (like \(3x^2\)).For example, in the expression \(2x^2 + 3x - 5\):

• \(2x^2\) is a term with a coefficient 2 and a variable \(x\) raised to the exponent 2.
• \(3x\) involves the coefficient 3 and variable \(x\) with an implicit exponent 1.
• e -5 is a constant term.

Algebraic expressions are pivotal in forming equations and functions, allowing us to represent complex mathematical ideas. They also provide a foundation for exploring more complex algebraic concepts and operations.