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Chapter 2: Problem 44

Simplify the given expression as completely as possible. $$m^{5} m^{4} m^{6}$$

Short Answer

Expert verified
The simplified expression is \(m^{15}\).

Step by step solution

01

- Identify the expression

Recognize the given expression: \(m^{5} m^{4} m^{6}\)
02

- Apply the Product of Powers rule

The Product of Powers rule states that when multiplying two exponents with the same base, you can add the exponents: \(a^n \times a^m = a^{n+m}\). Apply this rule to the bases of the given expression: \(m^{5} m^{4} m^{6}\).
03

- Add the exponents

Add the exponents together: \(5 + 4 + 6\). This results in 15. Therefore, \(m^{5} m^{4} m^{6} = m^{15}\).

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Key Concepts

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

Product of Powers Rule
When you multiply terms with the same base, you can simplify the expression using the Product of Powers rule. This rule states that when you multiply exponents with the same base, you simply add the exponents together.
For instance, the expression \(a^n \times a^m\) can be simplified to \(a^{n+m}\).
In our exercise, the base is \(m\), and we have three exponents: \(m^5\), \(m^4\), and \(m^6\). By applying the Product of Powers rule, we add the exponents together:
  • Step 1: Identify the exponents: 5, 4, and 6
  • Step 2: Add the exponents: 5 + 4 + 6 = 15
  • Step 3: Combine them: \(m^{15}\)
This simplifies the expression from \(m^5 m^4 m^6\) to \(m^{15}\).
Simplifying Exponents
Understanding how to simplify exponents is crucial in algebra. Exponents are shorthand for repeated multiplication. For example, \(m^5\) means \(m\) multiplied by itself five times: \(m \times m \times m \times m \times m\).
When simplifying exponents within the same base, the key is to use the rules like the Product of Powers rule. This rule helps make calculations easier. Suppose you have \(m^5 m^4 m^6\), it might seem complicated, but it's just a matter of combining the powers of the same base:
  • \(m^5 = m \times m \times m \times m \times m\)
  • \(m^4 = m \times m \times m \times m\)
  • \(m^6 = m \times m \times m \times m \times m \times m\)
Combining all these together gives \(m\) multiplied a total of 15 times, which is \(m^{15}\). Simplifying in this manner helps keep expressions neat and manageable.
Algebra Basics
Algebra is the branch of mathematics that uses letters to represent numbers in equations and formulas. These letters are known as variables.
The fundamental operations of algebra include adding, subtracting, multiplying, and dividing expressions with variables. In solving algebraic expressions, rules like the Product of Powers and properties of exponents play significant roles.
Key basics to remember:
  • Variables: Symbols like \(m\) representing numbers
  • Exponents: Shows how many times a number is multiplied by itself, e.g., \(m^5\)
  • Product of Powers: \(a^n \times a^m = a^{n+m}\)
Understanding these basics helps you tackle more complex problems efficiently. In our example, simplifying \(m^5 m^4 m^6\) leverages these foundational concepts, showing how algebra simplifies and resolves expressions clearly and effectively.

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