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Chapter 9: Problem 11

Multiply the following expressions. \(a^{6} \cdot a^{4} \cdot a^{2}\)

Short Answer

Expert verified
The product of the expressions is \(a^{12}\).

Step by step solution

01

Identify the Base and Exponents

In the given expressions, identify that the base is the same variable, which is \(a\). The exponents are 6, 4, and 2, respectively.
02

Apply the Law of Exponents for Multiplication

The product of powers rule states that when multiplying like bases, you add the exponents: \[ a^m imes a^n = a^{m+n}. \] We will apply this rule to the exponents 6, 4, and 2.
03

Add the Exponents

Add the exponents together: \(6 + 4 + 2 = 12\). Therefore, the expression becomes \(a^{12}\).
04

Write the Final Expression

Using the previous steps, the multiplied expression is \(a^{12}\). Ensure the final answer is fully simplified.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Multiplying Powers
Multiplying powers is an essential concept when working with exponents, especially when dealing with the same base. If you're multiplying expressions like \(a^6 \cdot a^4 \cdot a^2\), you're dealing with powers that need to be multiplied efficiently.
  • The first step is to recognize the common base in all expressions. For example, in this problem, the base is \(a\).
  • Next, look closely at the exponents. These are the numbers that tell you how many times the base is used in a multiplication.
  • With like bases, you simply add their exponents together - this law is due to the product of powers rule, which we'll discuss more in the next section.
By understanding the fundamental concept of multiplying powers, this exercise becomes much simpler to handle.
Product of Powers Rule
The product of powers rule is a key part of working with exponents. Let's break it down so you can apply it confidently.
  • The rule states that when you multiply powers with the same base, you add their exponents. Mathematically, it's expressed as \( a^m \cdot a^n = a^{m+n} \).
  • This simplifies the process significantly, allowing you to turn complex problems into manageable steps.
  • Remember, this rule only applies when the base is identical in each term. If the bases were different, you'd need a different approach.
By mastering the product of powers rule, you're well on your way to mastering all sorts of problems involving exponents.
Exponent Addition
When you encounter expressions like \(a^6 \cdot a^4 \cdot a^2\), adding exponents is a crucial part of simplifying the problem.
  • Start by listing the exponents involved: 6, 4, and 2 in this case.
  • Use addition to combine them: \(6 + 4 + 2 = 12\).
  • This step is straightforward but crucial for getting the right answer. It results in a much simpler final expression: \(a^{12}\).
Exponent addition enables you to overcome potential confusion when working with multiple powers. It forms a foundation for more complex operations in algebra.

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