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Chapter 5: Problem 27

Find each product. $$ 7 m^{3} n^{2}\left(3 m^{2}+2 m n-n^{3}\right) $$

Short Answer

Expert verified
21m^5n^2 + 14m^4n^3 - 7m^3n^5

Step by step solution

01

Distribute the First Term

Multiply the first term inside the parentheses by the outside term: \[ 7m^3n^2 \times 3m^2 \] This results in: \[ 21m^5n^2 \]
02

Distribute the Second Term

Now, multiply the second term inside the parentheses by the outside term: \[ 7m^3n^2 \times 2mn \] This results in: \[ 14m^4n^3 \]
03

Distribute the Third Term

Finally, multiply the third term inside the parentheses by the outside term: \[ 7m^3n^2 \times -n^3 \] This results in: \[ -7m^3n^5 \]
04

Combine the Results

Sum up all the results from steps 1 to 3: \[ 21m^5n^2 + 14m^4n^3 - 7m^3n^5 \]

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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 a crucial concept in algebra that allows you to simplify expressions. It states that for any numbers or expressions a, b, and c, the expression \(a(b + c)\) is equivalent to \(ab + ac\). It distributes multiplication over addition, making it easier to manage complex algebraic expressions.

In the given problem, we use the distributive property to multiply the term outside the parentheses by each term inside the parentheses:
  • First, multiply \(7m^3n^2\) by \(3m^2\).
  • Next, multiply \(7m^3n^2\) by \(2mn\).
  • Finally, multiply \(7m^3n^2\) by \(-n^3\).

This step-by-step multiplication illustrates how the distributive property works in practice.
Algebraic Expressions
An algebraic expression is a combination of variables, numbers, and at least one arithmetic operation. For example, \(3m^2 + 2mn - n^3\) is an algebraic expression. Understanding algebraic expressions is essential because they represent mathematical relationships and quantities.

In the exercise, the expression inside the parentheses \(3m^2 + 2mn - n^3\) includes terms with different variables and exponents. We multiply each of these terms by \(7m^3n^2\) to find the product. This process simplifies the expression and combines like terms effectively.
Exponents
Exponents indicate how many times a number or variable is multiplied by itself. For example, \(m^3\) means \(m \times m \times m\). When we multiply terms with exponents, we add the exponents if the bases are the same.

In the solution steps:
  • \[7m^3n^2 \times 3m^2 = 21m^5n^2\] because \((m^3 \times m^2) = m^{5}\).
  • \[7m^3n^2 \times 2mn = 14m^4n^3\] because \((m^3 \times m) = m^{4}\) and \((n^2 \times n) = n^{3}\).
  • \[7m^3n^2 \times -n^3 = -7m^3n^5\] because \((n^2 \times n^3) = n^{5}\).
This understanding of exponents simplifies the multiplication of algebraic expressions, making it easier to follow each step and arrive at the correct solution.

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Most popular questions from this chapter

Graph each equation by completing the table of values. \(y=x^{2}-6\) \(\begin{array}{|c|c|}\hline x & {y} \\ \hline-2 & {} \\ \hline-1 & {} \\\ \hline 0 & {} \\ \hline 1 & {} \\ \hline 2 & {} \\ \hline\end{array}\)

Perform each division. $$ \frac{4 t^{5}-11 t^{4}-6 t^{3}+5 t^{2}-t+3}{4 t^{2}+t-3} $$

Perform each indicated operation. Find the difference between the sum of \(5 x^{2}+2 x-3\) and \(x^{2}-8 x+2\) and the sum of \(7 x^{2}-3 x+6\) and \(-x^{2}+4 x-6\)

Simplify by writing each expression with positive exponents. Assume that all variables represent nonzero real numbers. See Example \(5 .\) $$ \frac{\left(m^{7} n\right)^{-2}}{m^{-4} n^{3}} $$

Perform each division. $$ \frac{x^{4}-4 x^{3}+5 x^{2}-3 x+2}{x^{2}+3} $$
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