Problem 45 - Algebra for College Students

Question

Simplify each expression. Express final results without using zero or negative integers as exponents. \(a^{3} \cdot a^{-5} \cdot a^{-1}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{a^3}\).

Step 1: Apply the Laws of Exponents

The first law of exponents to apply is that when multiplying expressions with the same base, you add the exponents. The expression is \(a^3 \cdot a^{-5} \cdot a^{-1}\).Applying the law results in: \[a^{3 + (-5) + (-1)}\] which simplifies to \(a^{-3}\).

Step 2: Convert Negative Exponent

To express the answer without a negative exponent, use the rule that \(a^{-n} = \frac{1}{a^n}\). Thus, \(a^{-3}\) becomes \(\frac{1}{a^3}\).

Key Concepts

Understanding Negative Exponents

Negative exponents can be a bit tricky at first, but they're actually just another way of expressing division. When you see a negative exponent, it means you're dealing with a reciprocal. For example, if you have an exponent like \(a^{-n}\), it can be rewritten as \(\frac{1}{a^n}\). This rule helps to transform expressions involving negative exponents into simpler forms.

Let's look at an example to deepen our understanding. If you come across \(x^{-2}\), it can be expressed as \(\frac{1}{x^2}\). Essentially, by taking the reciprocal, you flip the position of the base from numerator to denominator. This transformation is important when you aim to simplify expressions and eliminate negative exponents altogether.

Multiplying Exponents with the Same Base

Multiplying exponents that share the same base is a key concept in simplifying expressions. The trick is to add the exponents together when the bases are identical. Imagine you have something like \(b^m \cdot b^n\). According to the law of exponents, this can be simplified to \(b^{m+n}\).

Let’s break down an example: Consider \(a^3 \cdot a^{-5} \cdot a^{-1}\). All parts have the same base, \(a\). Using our rule, we need to add up all the exponents: \(3 + (-5) + (-1)\). Adding them gives us a resultant exponent of \(-3\). Hence, the expression simplifies to \(a^{-3}\).

This principle not only simplifies the problem but also lays the foundation for more advanced algebraic operations.

Simplifying Expressions Involving Exponents

Simplifying expressions involving exponents is all about using the laws of exponents strategically. The ultimate goal is to remove negative exponents and express everything in its simplest form. Once you've combined any like terms by multiplying exponents as we've previously discussed, it's time to handle any negative exponents.

For the expression \(a^{-3}\), applying what we learned about negative exponents, it becomes \(\frac{1}{a^3}\). This transformation keeps our expressions clean and easy to interpret.

• Always take care to properly adjust negative exponents by using reciprocals.
• Remember, the process of simplifying is not just about getting the right answer but making the expression as neat and interpretable as possible.
• Maintain clarity in each step to avoid errors and reinforce your understanding of exponent rules.

In practice, simplifying expressions helps you quickly identify patterns and relationships among terms, which is an invaluable skill in mathematics.