Problem 25 - College Algebra

Question

For the following exercises, simplify the given expression. Write answers with positive exponents. $$ \frac{a^{3} a^{2}}{a} $$

Step-by-Step Solution

Answer

The expression simplifies to \( a^4 \).

Step 1: Apply the Product of Powers Rule

The expression \( a^3 a^2 \) can be rewritten using the product of powers rule which states \( a^m imes a^n = a^{m+n} \). Therefore, \( a^3 a^2 = a^{3+2} = a^5 \).

Step 2: Divide Using the Quotient Rule

Now that you have \( a^5 \), apply the quotient of powers rule. This rule states \( \frac{a^m}{a^n} = a^{m-n} \). Substitute the powers into the rule: \( \frac{a^5}{a} = a^{5-1} = a^4 \).

Step 3: Final Simplified Expression

The expression \( \frac{a^3 a^2}{a} \) simplifies to \( a^4 \). Since \( a^4 \) already has a positive exponent, no further simplification is needed.

Key Concepts

Exponent Rules

Exponent rules are a set of guidelines that help us simplify expressions involving powers. These rules are essential when working with algebraic expressions that include repeated multiplication. Understanding exponent rules makes it easier to manipulate these expressions and find simplified forms. There are a few basic rules to remember:

• Power of a Power: When you raise a power to another power, multiply the exponents. For example, \((a^m)^n = a^{m \times n}\).
• Product of Powers: When multiplying like bases, add the exponents. This rule is used in the first step of the solution, \(a^m \cdot a^n = a^{m+n}\).
• Quotient of Powers: When dividing like bases, subtract the exponents. This rule simplifies the expression in step two of the solution, \(\frac{a^m}{a^n} = a^{m-n}\).
• Negative Exponents: They indicate reciprocals, but since this exercise requires positive exponents, negative exponents are avoided.

Getting familiar with these rules will make handling more complex algebraic expressions much easier and faster.

Product of Powers Rule

The product of powers rule is applied when you multiply two or more expressions with the same base. It's quite simple—just keep the base and add the exponents together. In our example, we started with \(a^3 \cdot a^2\). Both terms have the same base of \(a\), so we add their exponents: 3 plus 2. This gives us \(a^{3+2} = a^5\). This rule dramatically simplifies expressions, especially when dealing with multiple terms with the same base. By simply adding exponents, we reduce the complexity and arrive at a cleaner expression. The product of powers rule works for any number of terms with the same base, making it a versatile tool in algebra.

Quotient of Powers Rule

The quotient of powers rule is used when dividing expressions that have the same base. This rule means you take the exponent from the numerator and subtract the exponent in the denominator. For the exercise given, after applying the product of powers rule, we have \(\frac{a^5}{a}\). Here, it's important to note that the denominator simple has an implied exponent of 1 (i.e., \(a^1\)). So, by applying the rule \(\frac{a^m}{a^n} = a^{m-n}\), you get \(a^{5-1} = a^4\).This rule makes it easy to simplify complicated fractions quickly. By converting a division problem into a simple subtraction problem, we can quickly determine the expression with a reduced exponent. The quotient of powers rule ensures your final expression is easier to interpret and often cleaner to look at.