Problem 70 - Algebra 1: Concepts and Skills

Question

Simplify the expression \(5^{2} \cdot 5^{4}\) A) \(5^{6}\) B) \(5^{8}\) C) \(10^{6}\) D) \(25^{8}\)

Step-by-Step Solution

Answer

A) \(5^{6}\)

Step 1: Identifying Similar Bases and the Operation

The expression given is \(5^{2} \cdot 5^{4}\). Here, both the terms have the same base number, which is 5. Also, these terms are multiplied.

Step 2: Applying the Law of Exponents

When the bases are the same and the terms are being multiplied, add the exponents. Here, the expression becomes \(5^{(2+4)}\).

Step 3: Performing the Addition

Add the exponents together: \(5^{(2+4)} = 5^{6}\)

Key Concepts

Law of Exponents

Understanding the law of exponents is essential when working with exponential expressions. This law gives us a set of rules for performing operations on powers with the same or different bases. One of the key rules is that when you're multiplying exponents with the same base, you keep the base and add the exponents.

For example, the problem at hand deals with multiplying two powers of 5:

\(5^{2} \times 5^{4}\)

In accordance with the law of exponents, since the base (5) is the same, we can add the exponents. This simplifies the expression to:

\(5^{2+4}\)

By applying this law correctly, students can avoid common mistakes like multiplying the bases or the exponents incorrectly. It's helpful to remember that this law only applies when the bases are the same, as this is where some students may go wrong.

Exponential Notation

Exponential notation is a convenient way to represent repeated multiplication of the same factor. In an exponential expression, such as

\(a^{n}\)

the 'a' represents the base and the 'n' indicates the exponent, also known as the power. The exponent tells us how many times the base is multiplied by itself. For instance,

\(3^{4}\)

would mean 3 multiplied by itself 4 times (

\(3 \times 3 \times 3 \times 3\)

). When simplifying exponential expressions, it's necessary to understand this notation to follow the proper rules and perform calculations correctly.

Multiplying Exponents

When it comes to multiplying exponents, the process is straightforward if you remember one key rule: if the bases are the same, add the exponents. Let's take the given exercise as an instance. We multiply two expressions with an identical base of 5, namely:

\(5^{2}\times5^{4}\)

To multiply these, you simply need to add the exponents as per the law of exponents:

\(5^{2+4} = 5^{6}\)

It's important not to get confused and multiply the exponents themselves. The bases remain the same, and only the exponents are added. This method provides a concise and powerful tool for working with exponential numbers. Keeping this rule in mind helps streamline the simplification process, making it easier to solve even more complex problems involving exponents.