/*! 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 1 Multiply the following expressio... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 1

Multiply the following expressions. $$x^{2}\left(x^{4}+x^{3}\right)$$

Short Answer

Expert verified
The product of the expressions is \( x^{6} + x^{5} \).

Step by step solution

01

Distribute the Expression

To multiply the expression \( x^{2}(x^{4} + x^{3}) \), we start by distributing \( x^{2} \) to each term inside the parentheses. This means multiplying \( x^{2} \) by \( x^{4} \) and then \( x^{2} \) by \( x^{3} \).
02

Multiply the Terms

First, multiply \( x^{2} \) by \( x^{4} \). Using the rule of exponents which states that \( x^{a} \times x^{b} = x^{a+b} \), we get \( x^{2+4} = x^{6} \).Next, multiply \( x^{2} \) by \( x^{3} \). Again, applying the same rule gives us \( x^{2+3} = x^{5} \).
03

Combine the Results

Now, combine the results from the previous step. We have \( x^{6} \) from the first multiplication and \( x^{5} \) from the second. Thus, the expression after distribution and combining is: \[ x^{6} + x^{5} \]

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.

Understanding the Distributive Property
The distributive property is a fundamental principle used in algebra to simplify expressions and solve equations. It allows us to multiply a single term by each term inside a set of parentheses, breaking down complex expressions into manageable segments.

The distributive property can be expressed mathematically as:
  • If you have a term \( a \) being multiplied by a sum inside parentheses \((b + c)\), it can be distributed as \( a \times (b + c) = a \times b + a \times c \).
In the exercise \( x^{2}(x^{4} + x^{3}) \), we distribute \( x^{2} \) to each term inside the parentheses.

This breakdown simplifies the process and helps us manage each multiplication separately, illustrating why the distributive property is essential in algebraic manipulations.
Exploring the Laws of Exponents
The laws of exponents are crucial rules in algebra that govern how to handle expressions with exponents. These laws make it easier to work with powers and simplify expressions, especially when multiplying terms together. One of the key laws is:
  • \( x^{a} \times x^{b} = x^{a+b} \): When you multiply like bases, you add the exponents.
In the given exercise, this law is applied twice:
  • First, \( x^{2} \times x^{4} = x^{6} \) because you add the exponents: \( 2 + 4 \).
  • Second, \( x^{2} \times x^{3} = x^{5} \), adding the exponents: \( 2 + 3 \).
Understanding these fundamental rules helps simplify complex problems and ensures accuracy in multiplying powers efficiently, a skill essential for problem-solving in algebra.
Working with Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operations that represent a particular value or set of values. These expressions form the backbone of algebra, allowing for a wide range of mathematical manipulations.

In the context of polynomial multiplication, expressions like \( x^{2}(x^{4} + x^{3}) \) are simplified by breaking them down into smaller, easier-to-manage parts using the distributive property and laws of exponents.

Simplifying these expressions involves:
  • Identifying parts of the expression: constants, variables, and coefficients.
  • Applying the appropriate algebraic rules to combine like terms and simplify the expression.
By mastering these techniques, students can efficiently tackle more challenging algebra problems, building confidence in their mathematical skills and understanding.

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

Expand and multiply. $$(x+4)^{2}$$

Find the value of each of the following expressions when \(x=-3\). $$4 x+7$$

Simplify the following expressions. \(\left(9 x^{4}\right)^{2}\)

Expand and multiply. $$(x+6)^{2}$$

Multiply the following expressions. $$8 x^{2} y^{2}\left(5 x^{3}+4 y^{3}\right)$$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

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.