/*! 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 3 Use the properties of exponents ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 3

Use the properties of exponents to rewrite each expression. a. \(3 x^{5}(4 x)\) b. \(y^{8}\left(7 y^{8}\right)\) c. \(b^{4}\left(2 b^{2}+\mathrm{b}\right)\) d. \(2 x\left(5 x^{3}-3 x\right)\)

Short Answer

Expert verified
a. \(12x^6\); b. \(7y^{16}\); c. \(2b^6 + b^5\); d. \(10x^4 - 6x^2\).

Step by step solution

01

Distribute the Monomials

For each expression, we distribute the multiplication across the terms in the parentheses. This involves multiplying each term inside the parentheses by the term outside it. We start with statement (a), then (b), proceed to (c), and finally (d).
02

Simplify Using Exponent Rules

Apply the product of powers property which states that when you multiply like bases, you add the exponents. For example, for the expression \(a^m \times a^n\), we write it as \(a^{m+n}\).
03

Apply to (a)

Given expression is \(3x^5(4x)\). Distribute and multiply: \(3 imes 4 = 12\) and \(x^5 \times x^{1} = x^{5+1} = x^6\). Thus, the expression simplifies to \(12x^6\).
04

Apply to (b)

Given expression is \(y^8(7y^8)\). Distribute and multiply: \(7 \times y^8 \times y^8 = 7y^{8+8} = 7y^{16}\).
05

Apply to (c)

Given expression is \(b^4(2b^2 + b)\). Distribute \(b^4\) across terms: \(b^4 \times 2b^2 = 2b^{4+2} = 2b^6\) and \(b^4 \times b = b^{4+1} = b^5\). Therefore, the expression becomes \(2b^6 + b^5\).
06

Apply to (d)

Given expression is \(2x(5x^3 - 3x)\). Distribute \(2x\) across terms: \(2x \times 5x^3 = 10x^{1+3} = 10x^4\) and \(2x \times (-3x) = -6x^{1+1} = -6x^2\). Therefore, the expression becomes \(10x^4 - 6x^2\).

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.

Distributive Property
The distributive property is a fundamental concept in algebra that involves multiplying a single term by each term within a set of parentheses. This property simplifies expressions by spreading the multiplier across each term inside. Consider an expression like \(a(b+c)\). According to the distributive property, you can simplify this to \(ab + ac\). This property is especially helpful when dealing with polynomials, as it allows breaking down complex expressions for easier manipulation.
  • It's used to remove parentheses by distributing the multiplication.
  • This is often step one in simplifying any algebraic expressions.
Multiplication of Polynomials
When multiplying polynomials, such as a monomial by a binomial, each term in the polynomial must be multiplied by each term from the other polynomial. For instance, multiplying \(x(2x + 3)\) involves multiplying \(x\) by each term inside the parenthesis.
  • First, calculate \(x \times 2x = 2x^2\).
  • Then calculate \(x \times 3 = 3x\).
The result is the expanded polynomial: \(2x^2 + 3x\). This process is repeated for all types of polynomial expressions, which can include monomials, binomials, trinomials, and so forth.
Simplifying Expressions
Simplifying expressions in algebra means to make an expression as straightforward as possible. This involves both combining like terms and reducing expressions based on algebraic rules. When you simplify, you aim to have the cleanest form.
  • Use the distributive property to first distribute any multiplication over addition or subtraction.
  • Next, employ properties of exponents. For example, \(x^m \times x^n = x^{m+n}\).
  • Finally, combine any like terms to clean up the expression further.
By following these steps, you ensure that an expression is in its simplest, most efficient form.
Algebraic Expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. Unlike equations, algebraic expressions don't have an "equals" sign.
  • They can be as simple as a single variable like \(x\), or as complex as \(3x^2 + 5x - 9\).
  • When working with these expressions, you'll often simplify, factor, or expand them.
  • Mastering how to handle algebraic expressions involves understanding how to apply different algebraic principles like the distributive property and rules of exponents.
Clearly understanding how to manipulate these expressions is vital in algebra, as it lays the groundwork for solving equations and inequalities.

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

Write an equivalent expression in the form \(a \cdot b^{n}\). (hi) a. \(3 x \cdot 5 x^{3}\) b. \(x \cdot x^{5}\) c. \(2 x^{3}+2 x^{3}\) d. \(3.5(x+0.15)^{4} \cdot(x+0.15)^{2}\) e. \(\left(2 x^{3}\right)^{3}\) f. \(\left[3(x+0.05)^{3}\right]^{2}\)

Use the properties of exponents to rewrite each expression. Use your calculator to check that your expression is equivalent to the original expression. [ \(\square\) See Calculator Note 6B to learn how to check equivalent expressions. 4] a. \(3 x^{2} \cdot 2 x^{4}\) b. \(5 x^{2} y^{3} \cdot 4 x^{4} y^{5}\) c. \(2 x^{2} \cdot 3 x^{3} y^{4}\) d. \(x^{3} \cdot 4 x^{4}\)

APPLICATION Camila received a \(\$ 1,200\) prize for one of her essays. She decides to invest \(\$ 1,000\) of it for college. Her bank offers two options. The first is a regular savings account that pays \(2.5 \%\) interest every 6 months. The second is a certificate of deposit that pays \(5 \%\) interest each year. a. With the savings account, how much would Camila have after 1 year? After 2 years? (a) b. With the certificate of deposit, how much would Camila have after 1 year? After 2 years? c. Explain why you get different results for \(12 a\) and \(b\). (a)

On average a person sheds 1 million dead skin cells every 40 minutes. (The World in One Day, 1997, p. 16) a. How many dead skin cells does a person shed in an hour? Write your answer in scientific notation. (hi) b. How many dead skin cells does a person shed in a year? (Assume that there are 365 days in a year.) Write your answer in scientific notation.

Mini-Investigation In the last few lessons, you have worked with equations that have a variable exponent, and you have dealt with positive, negative, and zero exponents. An equation in which variables are raised only to nonnegative integer exponents is called a polynomial equation. Identify these equations as exponential, polynomial, or neither. (h) $$ \begin{array}{llll} y=4 x^{3} & y=-3(1+0.4)^{x} & y=x^{x}+x^{2} & y=2 x^{5}-3 x^{2}+4 x+2 \\ y=2 \cdot 3^{x} & y=2 x+7 & y=-6+2 x+3 x^{2} & y=3 \end{array} $$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

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.