Problem 29 - College Algebra

Question

Simplify each exponential expression in Exercises 23–64. $$x^{-5} \cdot x^{10}$$

Step-by-Step Solution

Answer

The simplified form of the expression \(x^{-5} \cdot x^{10}\) is \(x^{5}\).

Step 1: Understanding the Exponents

This problem has two parts: \(x^{-5}\) and \(x^{10}\). Here, 'x' is the common base, and '-5' and '10' are the exponents.

Step 2: Applying the Laws of Exponents

When multiplying two exponents with the same base, you add the exponents. In other words: \(x^m \cdot x^n = x^{m+n}\). In this case, m is -5 and n is 10 so: \(x^{-5} \cdot x^{10} = x^{-5+10}\).

Step 3: Solving the Expression

Simplify the exponent by adding -5 and 10 together: \(x^{-5+10} = x^{5}\).

Key Concepts

Laws of Exponents

Exponents are an essential part of algebra, and understanding their laws is crucial. The **laws of exponents** provide a framework for simplifying expressions with exponents with a common base. An important law is the product of powers rule. When multiplying exponents that share the same base, you add the powers together. For example, the expression \(x^m \cdot x^n\) simplifies to \(x^{m+n}\).

This means if you know how to add numbers, you can combine exponents when the base remains unchanged. Applying this law helps in reducing complex exponential expressions into simpler forms. Remember that these laws make exponential calculations straightforward and are foundational for more advanced mathematics.

Multiplying Exponents

Multiplying exponents is straightforward when following the laws of exponents. For expressions with a common base, you only need to add the exponents. Let’s look at an example: \(x^{-5} \cdot x^{10}\).

The base, \(x\), is the same in both terms, so according to our rule, you add the exponents \(-5\) and \(10\):

• Write the expression: \(x^{-5} \cdot x^{10}\).
• Add the exponents: \(-5 + 10\).
• Result: \(x^{5}\).

This process emphasizes the importance of the common base and shows how easily exponents can be combined using addition. This simplification is not only mathematically satisfying but also enhances your problem-solving efficiency in algebra.

Simplifying Exponents

Simplifying exponent expressions involves reducing them to their most basic form. The **exponential rules** assist in transforming complex expressions into simpler terms by combining powers when possible. In our example of \(x^{-5} \cdot x^{10}\), applying the laws of exponents quickly simplified the expression to \(x^{5}\) by adding the exponents.

This simplification removes any unnecessary complexity in calculations and helps understand the expression's actual value. One crucial concept in simplifying exponents is recognizing negative exponents. A negative exponent indicates the reciprocal of the base raised to the opposite positive power. However, once simplified using rules like "product of powers," negative exponents often resolve into positive ones, making calculations more straightforward.

• Combine exponents by addition when bases match.
• Remember that negative exponents signify reciprocal relationships.
• Always strive for the simplest form of the expression.

Understanding how to simplify exponents is essential for success in algebra and beyond.