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Chapter 3: Problem 6

For Problems \(1-36\), find each product. $$ \left(-8 a^{2} b^{2}\right)\left(-3 a b^{3}\right) $$

Short Answer

Expert verified
The product is \(24 a^3 b^5\).

Step by step solution

01

Identify the Expression

The given expression is \[(-8 a^2 b^2)(-3 a b^3)\]. We need to find the product of these two expressions.
02

Multiply the Coefficients

First, we multiply the coefficients of the terms \(-8\) and \(-3\).The product is:\[ -8 \times -3 = 24 \].
03

Multiply the "a" Terms

Next, we multiply the powers of \(a\) from both terms:\(a^2\) and \(a\).The result is:\[ a^2 \cdot a^1 = a^{2+1} = a^3 \].
04

Multiply the "b" Terms

Now, we multiply the powers of \(b\) from both terms:\(b^2\) and \(b^3\).The result is:\[ b^2 \cdot b^3 = b^{2+3} = b^5 \].
05

Write the Final Product

Combine the results from the steps:1. The coefficient is \(24\).2. The power of \(a\) is \(a^3\).3. The power of \(b\) is \(b^5\).Thus, the product is:\[ 24 a^3 b^5 \].

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Key Concepts

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

Product of Monomials
When you multiply monomials, you're looking at a really fundamental part of algebra. Monomials are algebraic expressions that contain just one term. In the exercise, the terms are \(-8a^2b^2\) and \(-3ab^3\). To find their product, you focus on two key aspects:
  • Coefficients: These are the numerical parts. Multiply them together just like ordinary numbers.
  • Variables with exponents: Multiply these by adding their exponents, respecting each variable separately.
Breaking it down:
  • The product of coefficients \(-8\) and \(-3\) is \(24\).
  • For the variables, multiply by adding the exponents of like terms.
Understanding these basics, you see how simple multiplying monomials really is. You just separate each part鈥攃oefficients and variables鈥攁nd handle them individually.
Exponent Rules
Exponents come into play when you multiply terms with the same base. The rule is straightforward: when you multiply, you add the exponents. The exercise involved two variables:
  • For the variable \(a\): You have \(a^2\) and \(a^1\). When multiplied, you get \(a^{2+1} = a^3\).
  • For the variable \(b\): You have \(b^2\) and \(b^3\). When multiplied, you get \(b^{2+3} = b^5\).
So why do we add exponents? Because multiplying a base by itself is equivalent to raising it to a new power. This rule is a cornerstone of working with algebraic expressions and helps simplify otherwise complex equations. As you apply these rules, you shrink the product of similar bases into a more compact form.
Algebraic Expressions
Algebraic expressions involve connecting numbers, variables, and operations. In our exercise, we dealt with the expression:\((-8a^2b^2)(-3ab^3)\).This expression included:
  • Two coefficients, \(-8\) and \(-3\), which were multiplied.
  • Variables \(a\) and \(b\), each raised to powers, which required applying exponent rules during multiplication.
Recognizing the types of components in algebraic expressions helps manage calculations:
  • Each part should be evaluated separately and then combined for the final result.
  • The simplicity of working with algebraic expressions hinges on maintaining clarity between coefficients and variable parts.
Such clarity is pivotal in comprehending more complex algebraic manipulations as you progress in mathematics.

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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) $$ 1-x^{2}=0 $$

For Problems \(1-54\), solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. (Objective 1) $$ 2 x^{2}+x-3=0 $$

Problems \(63-100\) should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. (Objective 4) $$ 2 x y+6 x+y+3 $$

For Problems \(1-54\), solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. (Objective 1) $$ 9 x^{4}-37 x^{2}+4=0 $$

Problems \(63-100\) should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. (Objective 4) $$ 25 t^{2}-100 $$
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