Problem 47 - College Algebra with Modeling and Visualization

Question

Exercises \(45-54:\) Use the quotient rule to simplify the expression. Use positive exponents to write your answer. $$ \frac{a^{-3}}{a^{2} \cdot a} $$

Step-by-Step Solution

Answer

\( \frac{1}{a^6} \)

Step 1: Identify Like Bases in the Denominator

The denominator of the expression is \(a^2 \cdot a\). These can be combined because they have the same base, \(a\).

Step 2: Combine Exponents in the Denominator

Apply the product rule for exponents in the denominator: add the exponents. This gives us \(a^{2+1} = a^3\). Now the expression becomes \(\frac{a^{-3}}{a^3}\).

Step 3: Apply the Quotient Rule for Exponents

The quotient rule states \( \frac{a^m}{a^n} = a^{m-n} \). Apply this rule: \( \frac{a^{-3}}{a^{3}} = a^{-3-3} = a^{-6}\).

Step 4: Rewrite with Positive Exponents

Since we want positive exponents, rewrite \(a^{-6}\) as \(\frac{1}{a^6}\).

Key Concepts

Exponents

Exponents are a compact way to represent repeated multiplication of the same number. When a number is raised to a power, it is multiplied by itself as many times as the exponent indicates. For example:

• If you see something like \( a^3 \), it means \( a \times a \times a = a^3 \).

There are also negative exponents, which essentially denote division. For example:

• \( a^{-3} \) is equivalent to \( \frac{1}{a^3} \).

Negative exponents do not mean the number itself is negative; they indicate the reciprocal of the base raised to the corresponding positive exponent.
Understanding how to manipulate exponents is an essential skill in algebra. It allows for simplifying complex expressions and solving equations more easily.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing them into their simplest form. This often means rewriting expressions in ways that make them easier to understand or evaluate. Let's break down the process from the exercise to showcases how simplifications are achieved.
First, we identified the like bases in algebraic expressions, such as \( a^2 \cdot a \). Since both terms have the same base, they can be combined using rules for exponents.

• In this case, you add the exponents together to simplify, transforming \( a^2 \cdot a \) into \( a^{3} \).

Once our denominator is simplified, we apply the quotient rule for exponents to deal with the fraction.
This rule allows us to subtract the exponents when dividing like bases, e.g., \( \frac{a^{-3}}{a^{3}} = a^{-3-3} = a^{-6} \).

• Finally, always convert any negative exponents into positive. \( a^{-6} \) becomes \( \frac{1}{a^6} \), making the expression more straightforward and consistent.

Product Rule for Exponents

The product rule for exponents is essential when you multiply expressions with the same base. It states that when multiplying like bases, you can add the exponents. For instance, with:

• \( a^m \times a^n = a^{m+n} \).

In our original exercise, we used the product rule to combine terms in the denominator:

• \( a^2 \times a = a^{2+1} = a^3 \).

By applying this rule, algebraic expressions are simplified into more manageable forms, as the total multiplication is represented by a single exponent.
Applying the product rule appropriately helps prevent errors and makes complex algebraic expressions much simpler to work with. This allows for easier integration of other rules and processes, such as the quotient or power rules, which depend on recognizing and simplifying your expressions correctly.