/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 Find each product or quotient. E... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 6

Find each product or quotient. Express using exponents. $$-3 x^{2}\left(4 x^{3}\right)$$

Short Answer

Expert verified
The product is \(-12x^{5}\).

Step by step solution

01

Identify the Components

Look at the expression \(-3 x^{2}(4 x^{3})\). The components consist of coefficients \(-3\) and \(4\), and the variables with exponents \(x^{2}\) and \(x^{3}\). Our goal is to find the product of these parts.
02

Multiply the Coefficients

First, multiply the coefficents \(-3\) and \(4\): \(-3 \times 4 = -12\).
03

Multiply the Variables Using Exponent Rules

Apply the rule of exponents which states that when you multiply like bases, you add the exponents \(a^{m} \times a^{n} = a^{m+n}\). Here, multiply \(x^{2}\) and \(x^{3}\): \(x^{2+3} = x^{5}\).
04

Form the Final Expression

Combine the results from Step 2 and Step 3. The product of the expression is now \(-12x^{5}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Multiplication of Coefficients
When dealing with algebraic expressions like \(-3 x^{2}(4 x^{3})\), it's crucial to first understand how to handle the coefficients. Coefficients are the numerical parts of terms, separate from the variables. Here, \(-3\) and \(4\) are the coefficients. Multiplying coefficients involves basic arithmetic: you simply multiply the numerical values together.

In this case, you multiply \(-3\) by \(4\) to get \(-12\). The negative sign is important too; remember a negative multiplied by a positive is always negative. Therefore, this step simplifies the part of the expression with numbers.

  • Identify the numerical coefficients.
  • Multiply them directly: \(-3 \times 4 = -12\).
This step deals only with numbers, setting the stage for handling the variable part of the problem.
Exponents Addition
When multiplying expressions with the same base, exponents play a critical role. The rule for multiplying exponents is simple. If you have \(x^{m} \times x^{n}\), then you add the exponents: \(x^{m+n}\).

In our example, \(x^{2}\) and \(x^{3}\) are the variable parts. Since they share the same base, \(x\), you add the exponents together: \(x^{2+3} = x^{5}\). This application of the exponent rule makes managing the algebraic expression much simpler.

  • Ensure the bases are the same before applying the rule.
  • Add the exponents: \(2 + 3 = 5\).
By systematically adding exponents, you maintain the integrity of the mathematical operation.
Algebraic Expressions
Combining the multiplication of coefficients and the addition of exponents results in an expression known as an algebraic expression. An algebraic expression can contain numbers, variables, and exponents.

Our task was to simplify: \(-3 x^{2}(4 x^{3})\). After applying the multiplication of coefficients and the addition of exponents, the final expression becomes \(-12x^{5}\).

  • Start by simplifying individual components.
  • Combine the results to form a cohesive expression.
This process of simplification takes expressions that may initially seem complex and translates them into manageable, straightforward forms. By understanding each component's role, the wider algebraic operation comes into focus, reinforcing your grasp on foundational algebraic principles.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use the Distributive Property to rewrite each expression. $$(n-3) 5$$

For each expression, use parentheses to group the numbers together and to group the powers with like bases together. Example: \(a \cdot 4 \cdot a^{3} \cdot 2=(4 \cdot 2)\left(a \cdot a^{3}\right)\) $$3 \cdot x^{4} \cdot(-5) \cdot x^{2}$$

Numbers can also be expressed in expanded form. Example \(1: 13,548=10,000+3000+500+40+8\) \(=\left(1 \times 10^{4}\right)+\left(3 \times 10^{3}\right)+\left(5 \times 10^{2}\right)+\left(4 \times 10^{1}\right)+\left(8 \times 10^{0}\right)\) Example \(2: 0.568=0.5+0.06+0.008\) $$ =\left(5 \times 10^{-1}\right)+\left(6 \times 10^{-2}\right)+\left(8 \times 10^{-3}\right) $$ Write each number in expanded form. $$5931$$

For each expression, use parentheses to group the numbers together and to group the powers with like bases together. Example: \(a \cdot 4 \cdot a^{3} \cdot 2=(4 \cdot 2)\left(a \cdot a^{3}\right)\) $$s \cdot t^{2} \cdot s \cdot t$$

Simplify each fraction. If the fraction is already in simplest form, write simplified. $$\frac{4 t}{64 t^{2}}$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.