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Chapter 7: Problem 11

Simplify each of the following as completely as possible. $$m^{6} m^{4} m^{3}$$

Short Answer

Expert verified
The simplified expression is \(m^{13}\).

Step by step solution

01

Identify the problem

The expression given is a multiplication of powers with the same base: \[m^{6} m^{4} m^{3}\]
02

Apply the Law of Exponents

Use the product of powers property which states: \[a^m \times a^n \times a^p = a^{m+n+p}\] For the given problem, this means:\[m^{6+4+3}\]
03

Add the Exponents

Now, add the exponents together:\[6 + 4 + 3 = 13\]Therefore, the expression simplifies to:\[m^{13}\]

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Key Concepts

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

multiplication of powers
When you come across an expression like \[m^{6} \times m^{4} \times m^{3}\]it's important to identify that you are dealing with multiplication of powers that have the same base. The base here is \(m\). In multiplying powers, each of the exponents represents repeated multiplication of the base. For instance, \(m^6\) means \(m \times m \times m \times m \times m \times m\). Similarly, \(m^4\) means \(m \times m \times m \times m\)and \(m^3\) means \(m\times m \times m\). By identifying that the bases are the same, we know we can combine the exponents into a more simplified form.
product of powers property
Now, let's take a look at the 'product of powers property', which is a fundamental part of the law of exponents. The property states:\[a^m \times a^n \times a^p = a^{m+n+p}\].This rule allows us to combine exponents when the bases are the same. Essentially, rather than multiplying the terms of the base multiple times, we simply add the exponents together, making calculations much simpler and quicker.In our example, we have three exponents with the same base \(m\), so using the product of powers property, we can rewrite the expression as \(m^{6+4+3}\).
simplification of expressions
To simplify an expression, we combine the exponents by performing the necessary arithmetic operation. For the given expression:\[m^{6+4+3}\],we add up the exponents: \(6 + 4 + 3 = 13\).So, the expression simplifies to \(m^{13}\). This means that multiplying \(m^{6}\), \(m^{4}\), and \(m^{3}\) is the same as raising \(m\) to the power of \(13\). Simplifying expressions like this reduces complexity and makes calculations easier to manage!

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Most popular questions from this chapter

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