Problem 82 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Product

The expression given is \((x^{3n-4})(x^4)\). To simplify, we need to apply the laws of exponents to multiply these terms.

Step 2: Apply the Law of Exponents

Recall the law of exponents: when multiplying like bases, add the exponents. Thus, the expression becomes \(x^{3n-4+4}\).

Step 3: Simplify the Exponent

Combine the exponents in the expression \(3n - 4 + 4\), which simplifies to \(3n\). Thus, the expression becomes \(x^{3n}\).

Step 4: Write the Final Product

After simplifying, the product of \((x^{3n-4})(x^4)\) is \(x^{3n}\).

Key Concepts

Laws of Exponents

The laws of exponents form the foundation for simplifying expressions involving powers. One of the most important laws is the Product of Powers Property, which states that when you multiply like bases, you should add their exponents. This can be expressed mathematically as:

• \( a^m \times a^n = a^{m+n} \)

In this law, \( a \) is the common base, and \( m \) and \( n \) are the exponents. This rule makes calculations easier, allowing you to work directly with the exponents rather than multiplying full numbers repeatedly.
For example, in the expression \( x^{3n-4} \times x^4 \), both parts have the base \( x \). According to the law, you add the exponents: \((3n-4) + 4\). Understanding this law is crucial in math, making it simpler to deal with expressions containing powers.

Multiplying Variables

When multiplying variables with exponents, it's essential to recognize their common bases. This commonality is key to efficiently applying exponents' laws. The process involves a straightforward step-by-step approach:

• Identify variables that share the same base.
• Apply the rule: add the exponents of these like bases.
• Simplify by combining the exponents.

This method can be applied to any expression with consistently based variables. For instance, in \( (x^{3n-4})(x^4) \), we first look at the variable \( x \) in both parts. Since they are the same, we apply the rule for exponents, adding them together. This ultimately simplifies the expression wholly into a single power of \( x \), completing the multiplication.

Simplifying Exponents

After applying the laws of exponents, the next critical step is simplifying the resultant expression. This is where basic arithmetic and algebra come into play. To simplify exponents effectively:

• Execute any addition or subtraction within the exponents themselves.
• Ensure the expression is reduced to its simplest form by resolving any internal expressions.
• Check that all like terms have been combined into one cohesive expression.

For the problem \( x^{3n-4+4} \), simplifying involves calculating \( 3n-4+4 \). Significantly, \( -4 \) and \( 4 \) cancel each other out, reducing the expression to \( 3n \). Consequently, the expression simplifies neatly to \( x^{3n} \). This step helps in attaining a cleaner, more manageable form of the result, allowing for easier interpretation and further calculations.