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Chapter 3: Problem 2

For Problems \(1-36\), find each product. $$ \left(6 x^{3}\right)\left(7 x^{2}\right) $$

Short Answer

Expert verified
The product is \(42x^5\).

Step by step solution

01

Multiply the Coefficients

To begin finding the product, first multiply the coefficients (numerical factors) of the two terms. Here the coefficients are 6 and 7. Calculate \(6 \times 7\) to get 42.
02

Apply the Law of Exponents

Next, you need to multiply the variables with exponents. According to the law of exponents, when you multiply like bases, you add the exponents. Here, multiply \(x^3\) and \(x^2\) which results in \(x^{3+2} = x^5\).
03

Combine the Results

Combine the results from Steps 1 and 2. The coefficient 42 and the exponentiated base \(x^5\) combine to give the final product: \(42x^5\).

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Key Concepts

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

Law of Exponents
The law of exponents is a set of rules that governs the operations on numbers with exponents. When multiplying powers with the same base, you simply add the exponents together. For example, if you have \(x^3\) and \(x^2\), these share the base "x". According to this rule, their product is \(x^{3+2} = x^5\).

This concept is crucial because it simplifies calculations involving powers. Instead of expanding the expressions, you can apply this easy rule to reach a solution quickly. Exponents indicate how many times a number should be multiplied by itself, so this rule helps manage potentially large numbers efficiently. Further understanding this rule can make complex algebraic operations more intuitive.
Coefficients
Coefficients are the constant numbers that are multiplied by the variables in algebraic expressions. In the product \((6x^3)(7x^2)\), 6 and 7 are the coefficients. These are multiplied directly to calculate the numerical part of the product. So, multiplying \(6 \times 7\) gives 42.

Notice the coefficients operate independently of the exponents. They simplify the process of multiplication by providing the numerical outcome, while variables handle the algebraic computations separately. Understanding coefficients helps in applying the multiplication steps in the right sequence and executing them confidently.
Exponents
Exponents are shorthand for expressing repeated multiplication of a number by itself. They consist of a base (the number being multiplied) and an exponent (how many times the base is used as a factor). For instance, in \(x^3\), the base is \(x\) and the exponent is 3, which means \(x\) is used as a factor three times.

Recognizing the behavior of exponents is essential for operations like multiplication and division. Using the law of exponents can simplify expressions. When you see terms like \(x^3\) and \(x^2\), it implies multiplying \(x\) three times and then twice, which, by the rule, becomes \(x^{3+2} = x^5\).
Learning about exponents enhances your ability to deal with powers and simplifies complex expressions into more manageable forms.

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Most popular questions from this chapter

For Problems \(1-54\), solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. (Objective 1) $$ 16-x^{2}=0 $$

Consider the following approach to factoring the problem $$\begin{aligned} &(x-2)^{2}+3(x-2)-10 \\ &(x-2)^{2}+3(x-2)-10 \\ &=y^{2}+3 y-10 \\ &=(y+5)(y-2) \\ &=(x-2+5)(x-2-2) \\ &=(x+3)(x-4) \end{aligned}$$ Use this approach to factor Problems \(110-115\). $$ 15(x+2)^{2}-13(x+2)+2 $$

For Problems \(1-54\), solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. (Objective 1) $$ (x+8)(x-6)=-24 $$

Consider the following approach to factoring the problem $$\begin{aligned} &(x-2)^{2}+3(x-2)-10 \\ &(x-2)^{2}+3(x-2)-10 \\ &=y^{2}+3 y-10 \\ &=(y+5)(y-2) \\ &=(x-2+5)(x-2-2) \\ &=(x+3)(x-4) \end{aligned}$$ Use this approach to factor Problems \(110-115\). $$ 6(x-4)^{2}+7(x-4)-3 $$

For Problems \(1-54\), solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. (Objective 1) $$ n(n+1)=182 $$
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