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Chapter 4: Problem 26

Find each product. $$ 2 m^{4}\left(6+5 m+3 m^{2}\right) $$

Short Answer

Expert verified
12 m^{4} + 10 m^{5} + 6 m^{6}

Step by step solution

01

Distribute the first term

Multiply the first term inside the parentheses \(6\) by \(2 m^{4}\). This gives: \(2 m^{4} \times 6 = 12 m^{4}\).
02

Distribute the second term

Multiply the second term inside the parentheses \(5 m\) by \(2 m^{4}\). This gives: \(2 m^{4} \times 5 m = 10 m^{5}\).
03

Distribute the third term

Multiply the third term inside the parentheses \(3 m^{2}\) by \(2 m^{4}\). This gives: \(2 m^{4} \times 3 m^{2} = 6 m^{6}\).
04

Combine all terms

Add all the products obtained in the previous steps together: \(12 m^{4} + 10 m^{5} + 6 m^{6}\).

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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 foundational concept in algebra that allows you to simplify expressions. It states that multiplying a sum by a number is the same as multiplying each addendum by the number and then adding the products. This concept can be written as: \( a(b + c) = ab + ac \).

In the provided exercise, we applied the distributive property to multiply each term inside the parentheses by the factor outside, which was \(2m^{4}\).

This allowed us to break the expression down into simpler parts, making it easier to solve. Remembering how to correctly apply the distributive property is crucial in solving more complex math problems.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and arithmetic operations (addition, subtraction, multiplication, division). Understanding these expressions is key to solving algebra problems.

The given exercise asked us to simplify an algebraic expression involving multiplication and addition of terms. The expression was \(2m^{4}(6 + 5m + 3m^{2})\).

By identifying and correctly manipulating the terms, we could solve the problem step-by-step. Always look for ways to simplify expressions, such as combining like terms or using properties like the distributive property.
Exponents
Exponents indicate how many times a number (the base) is multiplied by itself. In the exercise, we worked with terms like \(m^{4}\) and \(m^{2}\), which are examples of exponents. When multiplying terms with the same base, we add the exponents. For example, \(m^{4} \times m^{2} = m^{6}\).

This property is very useful and was a critical part of our solution.

Remembering the rules of exponents can help you simplify expressions and solve problems more efficiently. For example:
  • \textbf{Product of Powers Rule}: \(a^{m} \times a^{n} = a^{(m+n)}\)
  • \textbf{Power of a Power Rule}: \( (a^{m})^{n} = a^{(m \times n)} \)
  • \textbf{Power of a Product Rule}: \( (ab)^{m} = a^{m} \times b^{m} \)

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Most popular questions from this chapter

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ (x+7)^{2} $$

Find each product. $$ (4 x+3)(2 y-1) $$

Graph each equation by completing the table of values. $$\begin{aligned} &y=x^{2}-9\\\&\begin{array}{c|c}\hline x & y \\\\\hline-2 & \\\\\hline-1 & \\\\\hline 0 & \\\\\hline 1 & \\\\\hline 2 &\end{array}\end{aligned}$$ $$

Perform each indicated operation. Subtract \(-5 w^{3}+5 w^{2}-7\) from \(6 w^{3}+8 w+5\)

Perform each indicated operation. \(\left(16 x^{3}-x^{2}+3 x\right)+\left(-12 x^{3}+3 x^{2}+2 x\right)\)
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