Problem 81 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The product is \( a^{5n+1} \).

Step 1: Identify the Powers

We are given the expression \( (a^{5n-2})(a^{3}) \). In this expression, the base \( a \) is common in both terms. This allows us to combine the exponents according to the rule for multiplying powers with the same base.

Step 2: Apply the Product of Powers Rule

The rule for multiplying powers with the same base states that you add the exponents: \( a^m \times a^n = a^{m+n} \). Apply this rule to the expression:\[ (a^{5n-2})(a^3) = a^{(5n-2)+3} \].

Step 3: Simplify the Exponents

Simplify the exponent by performing the addition \((5n-2) + 3 \): \[ 5n - 2 + 3 = 5n + 1 \].

Step 4: Write the Final Expression

After simplifying, the expression becomes: \[ a^{5n + 1} \].

Key Concepts

Product of Powers Rule

The Product of Powers Rule is a fundamental concept in exponentiation that simplifies mathematical expressions involving the multiplication of numbers with the same base. When we have an expression like \(a^m \ imes a^n\), we can simplify it using this rule. The rule tells us that to multiply powers with the same base, we add their exponents: \(a^m \ imes a^n = a^{m+n}\).

This rule helps streamline calculations and is pivotal when working with algebraic expressions. For instance, consider our previous example where \(a^{5n-2} \ imes a^3\) is given. By applying the Product of Powers Rule, we add the exponents. This simplifies the operation to a single step: sum the exponents \((5n-2) + 3\) and express the result as \(a^{5n+1}\).

The beauty of this rule lies in its simplicity and efficiency. Next time you encounter similar expressions, remember, it's as simple as adding the exponents while maintaining the same base.

Exponent Rules

Exponent rules are essential tools in mathematics that allow us to simplify expressions involving powers and exponents. These rules are universally applicable, meaning they hold true regardless of the specific numbers involved.

Here are some key exponent rules to remember:

• Product of Powers Rule: \(a^m \ imes a^n = a^{m+n}\).
• Quotient of Powers Rule: This rule states \(a^m \div a^n = a^{m-n}\) when \m \geq n\.
• Power of a Power Rule: \( (a^m)^n = a^{m \times n}\).
• Zero Exponent Rule: Any base raised to the zero power is 1, i.e., \(a^0 = 1\), assuming \a\ is not zero.

Understanding these rules is crucial for simplifying more complex algebraic expressions and solving exponential equations efficiently. For example, knowing that multiplying powers just involves adding exponents saves time and reduces errors during calculations.

In addition to simplifying expressions, exponent rules are foundational for higher-level mathematics topics, including calculus and discrete mathematics.

Simplifying Expressions

Simplifying expressions involves the process of condensing a mathematical expression into its simplest form. This often means reducing the number of terms or combining like terms to make the expression more manageable and easier to work with.

Let's apply this concept to the expression \(a^{5n-2} \ imes a^3\). Initially, this expression looks complex, but remember, the goal is to simplify it. By recognizing that both terms share the same base, you can use the Product of Powers Rule to combine them: \((a^{5n-2})(a^3) = a^{(5n-2) + 3}\).

Now, let's focus on simplifying the exponent itself: \(5n - 2 + 3\). By performing basic addition, we resolve this to \(5n + 1\). Thus, our expression simplifies neatly to \(a^{5n+1}\).

Simplifying expressions is not just about performing operations. It's about recognizing patterns and utilizing mathematical rules efficiently to present the simplest possible form. This skill is particularly valuable when solving equations or when engaged in more advanced mathematical problem-solving.