Problem 20 - Algebra 1: Concepts and Skills

Question

Write the expression as a single power of the base. \(8^{9} \cdot 8^{5}\)

Step-by-Step Solution

Answer

The expression \(8^{9} \cdot 8^{5}\) can be written as a single power of the base as \(8^{14}\).

Step 1: Identify the base and exponents

In the given expression, \(8^{9} \cdot 8^{5}\), the base for both terms is 8 and the exponents are 9 and 5 respectively.

Step 2: Apply the laws of exponents

When you multiply two terms with the same base, you can add the exponents. Therefore, the result will be the base, 8, raised to the power of the sum of the original exponents (9+5).

Step 3: Perform the addition

Adding 9 and 5 gives 14. Therefore, \(8^{9} \cdot 8^{5}\) simplifies to \(8^{14}\).

Key Concepts

Exponential Expressions

Understanding exponential expressions is essential for mastering various mathematical concepts. An exponential expression consists of a base and an exponent, where the base represents the number being multiplied by itself and the exponent denotes how many times it is multiplied. For instance, in the expression \(8^{9}\), 8 is the base and 9 is the exponent, which means 8 is multiplied by itself 9 times. These expressions can become complex quickly, so it's crucial to know how to handle and simplify them for easier computation and understanding.Simplification of exponential expressions is often made possible through the application of the laws of exponents. These rules make it straightforward to manage operations such as multiplication, division, and power raising when dealing with exponential expressions. With proper knowledge and application of these laws, complex expressions can be broken down into more manageable forms.

Simplifying Expressions

Simplifying expressions is a central concept in algebra that involves reducing a complex expression to its simplest form. This doesn't change the value of the expression; it merely makes it cleaner and often easier to work with. In working with exponents, for example, simplifying can involve reducing the expression to a single exponentiated term.To achieve this, one must be familiar with different rules and properties of arithmetic, such as combining like terms, factoring, using the distributive property, and importantly, applying laws of exponents. When it comes to exponential expressions, being efficient at simplifying can save considerable time and effort, and minimize mistakes in more advanced calculations. Moreover, in simplifying expressions like \(8^{9} \cdot 8^{5}\), we recognize both terms have the same base and employ the law of exponents stating that when multiplying powers with the same base, one can simply add the exponents. Through simplification, \(8^{9} \cdot 8^{5}\) becomes \(8^{14}\), a much neater and concise representation.

Multiplying Powers with the Same Base

When you encounter expressions where you are multiplying powers that share the same base, there's a beneficial shortcut to find the product. Instead of multiplying the bases multiple times as per the exponents, the law of exponents allows you to simply add the exponents together. This law streamlines calculations and reduces the potential for error in lengthy computations.For the multiplication of \(8^{9}\) and \(8^{5}\), both share the base of 8. By the law of exponents, you can add the exponents together, giving us \(8^{9+5}\) or \(8^{14}\), thus significantly simplifying the multiplication process. This not only simplifies the terms but also emphasizes the power of exponents in representing substantial numbers without extensive computation. Learning to quickly apply this rule when multiplying powers with the same base is a fundamental skill in algebra that aids in the simplification of expressions and in solving more complex equations.