Problem 30 - College Algebra

Question

Simplify each exponential expression in Exercises 23–64. $$x^{-6} \cdot x^{12}$$

Step-by-Step Solution

Answer

\(x^{6}\)

Step 1: Recognize the Negative Exponent Rule

A negative exponent indicates the reciprocal of that base. Hence \(x^{-6}\) can be rewritten as \(1/x^6\). So, \(x^{-6} \cdot x^{12}\) is equivalent to \(1/x^6 \cdot x^{12}\).

Step 2: Apply the Exponent Rule for Multiplication

When bases are the same, exponents add up during multiplication. Therefore, we sum up the exponents of \(x\) in \(1/x^6 \cdot x^{12}\). Effectively, we subtract 6 from 12 because \(1/x^6\) is the same as \(x^{-6}\). That simplifies the expression to \(x^{12 - 6}\).

Step 3: Simplify the Exponent

We perform the subtraction in the exponent, resulting in \(x^{6}\).

Key Concepts

Negative Exponents

Understanding negative exponents is crucial when dealing with exponential expressions. A negative exponent tells us that the base of the power should be taken as the reciprocal. Here's a simple way to think about it:
The expression \(x^{-a}\) is equivalent to \(\frac{1}{x^a}\). This means if you encounter a negative exponent, you should rearrange it by taking the reciprocal of the base.

• For example, \(x^{-3}\) becomes \(\frac{1}{x^3}\).
• With numbers, \(2^{-2}\) equals \(\frac{1}{2^2} = \frac{1}{4}\).

Recognizing and correctly rewriting negative exponents can often simplify more complicated algebraic problems. It helps to keep the operations clear and manageable.

Multiplication of Exponents

When multiplying exponential terms with the same base, the rule is to add the exponents. This concept applies even if you're dealing with negative exponents.
Let's look at how it works:

• If you have \(x^m \cdot x^n\), you will add the exponents to get \(x^{m+n}\).
• When including negative exponents, consider \(x^{-a} \cdot x^b\). Here, you add \(-a\) and \(b\), so it becomes \(x^{b-a}\).

In our example from the exercise, \(x^{-6}\) is multiplied by \(x^{12}\). Applying the multiplication of exponents rule means you perform the operation: \(-6 + 12 = 6\), leading to the simplified version of \(x^6\).
This rule significantly decreases the complexity of problems involving multiple exponential terms.

Simplifying Exponents

Simplifying exponents involves reducing expressions to their simplest form. The goal is to make expressions easier to understand and work with. After applying rules like those for negative exponents and multiplication, you often end up with a more manageable expression.

• For instance, after adding exponents in a product, you may end up with terms like \(x^6\), which is already simple.
• Consider combining terms beforehand, checking for opportunities to simplify at each stage. This ensures you're left with the clearest form.

The original problem \(x^{-6} \cdot x^{12}\) simplifies through these processes to \(x^6\). No fractions or additional operations are involved, just a neat, single expression. This demonstrates the power of truly understanding exponent rules.