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Chapter 13: Problem 19

Find each product. $$4 m\left(m^{2}-m\right)$$

Short Answer

Expert verified
The expanded expression is \(4m^3 - 4m^2\).

Step by step solution

01

Distribute the Terms

Apply the distributive property of multiplication over subtraction to expand the expression. Multiply 4m by each term inside the parentheses: \[ 4m(m^2 - m) = 4m \cdot m^2 - 4m \cdot m \].
02

Simplify Each Product

Simplify each product by applying the basic rules of exponents: \[ 4m \cdot m^2 = 4m^{1+2} = 4m^3 \] and \[ 4m \cdot m = 4m^{1+1} = 4m^2 \]. The expression becomes \[ 4m^3 - 4m^2 \].
03

Write the Final Expanded Expression

The expanded expression, after simplifying, is: \[ 4m^3 - 4m^2 \].

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Key Concepts

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

Distributive Property
The distributive property is an essential concept in algebra. It allows you to multiply a single term by each term within a parenthesis. This operation is particularly useful because it simplifies complex expressions and makes them easier to handle. In our example, we applied the distributive property to the expression \(4m(m^2 - m)\). This meant multiplying \(4m\) by each term inside the parentheses, resulting in \(4m \cdot m^2 - 4m \cdot m\).

By distributing the multiplication over subtraction (or addition), you effectively eliminate the parentheses, which simplifies the expression.
  • Step-by-step: Multiply the term outside by each term inside the parentheses.
  • Remember, the sign before each term inside the parentheses stays with the respective term.
This method turns products into sums or differences, which are easier to compute.
Simplification of Expressions
Simplifying expressions is about reducing them to their simplest form. It involves combining like terms and performing basic arithmetic operations. After distributing in the given expression, we had \(4m \cdot m^2\) and \(4m \cdot m\).

Both these products needed simplification. Here, simplifying involved applying exponent rules to combine the terms correctly. We combined the terms by adding the exponents of like bases, making calculations manageable.
  • The goal: Make the expression as straightforward as possible.
  • Combine terms so that each exponent and coefficient stands clearly.
The solution ending in \(4m^3 - 4m^2\) is a testament to effective simplification.
Exponents Rules
Exponents are shorthand for repeated multiplication of a number by itself. Understanding the rules governing exponents is crucial for simplifying algebraic expressions correctly. In our example, we utilized two key exponent rules:
  • Multiplying like bases: When you multiply terms with the same base, you add the exponents together. Thus, \(m^1 \cdot m^2\) becomes \(m^{1+2} = m^3\).
  • Same base multiplication: For \(4m \cdot m\), the exponents add up to \(m^{1+1} = m^2\).

Applying these rules enables you to express the product of like terms in a simpler form, making calculations more intuitive and less error-prone.
The expression \(4m^3 - 4m^2\) came to life through consistent application of these exponent rules, turning complex multiplication into a neat subtraction, which is easier to understand.

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Most popular questions from this chapter

The graph of quadratic functions may have one maximum or one minimum point. The maximum point of a graph is the point with the greatest \(y\) -value coordinate. The minimum point is the point with the least \(y\) -value coordinate. Graph each equation. Find the coordinates of each point. the maximum point of the graph of \(y=-x^{2}+7\)

Express each percent as a decimal. $$421 \%$$

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