/*! 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 15 Simplify each expression using t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 15

Simplify each expression using the products-to-powers rule. $$(2 x)^{3}$$

Short Answer

Expert verified
The simplified form of the expression \((2x)^3\) is \(8x^3\).

Step by step solution

01

Identify the Base and Exponent

In the given expression \((2x)^3\), the base is '2x' and the exponent is '3'.
02

Apply the Products-to-Powers Rule

By applying the products-to-powers rule \((ab)^n = a^nb^n\), the expression becomes \(2^3 * x^3\).
03

Simplify the Expression

Upon simplifying \(2^3\), which equals to '8', the final expression becomes \(8x^3\).

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.

Exponentiation
Exponentiation is a fundamental operation in mathematics. It involves taking a number, called the base, and raising it to a power, which is the exponent. This operation tells you how many times to use the base in a multiplication. For instance, if you have the expression \( (2x)^3 \), it means that the term \(2x\) is multiplied by itself three times.
Think of exponentiation like repeatedly multiplying the same number:
  • \( a^2 \) means \( a \times a \)
  • \( a^3 \) means \( a \times a \times a \)
  • \( a^4 \) means \( a \times a \times a \times a \)
Understanding exponentiation is key because it simplifies repeated multiplications into a more usable form. It saves space in writing and makes many calculations less cumbersome.
Base and Exponent
In expressions involving powers, you'll always find a base and an exponent. The base is the number or expression being multiplied. The exponent, also sometimes called the power, tells you how many times to use the base as a factor in the multiplication.
Let's break it down with the given example \((2x)^3\):
  • "2x" is the base. It represents what is being repeatedly multiplied.
  • "3" is the exponent. It indicates that \(2x\) should be used in multiplication three times: \(2x \times 2x \times 2x\).
When combined, the base and exponent form what鈥檚 known as a "power". It's important to recognize this relationship because it dictates how you apply the power rules, such as the products-to-powers rule, which can simplify complex expressions.
Simplifying Expressions
Simplifying expressions often involves using certain rules to make the expression easier to work with or understand. For the example \((2x)^3\), you can simplify it using the products-to-powers rule. This rule states that if you have a product inside a power, you can expand it to apply the power to each term in the product:
  • \((ab)^n = a^n b^n\)
Using this rule, the expression \((2x)^3\) becomes \(2^3 \times x^3\). This lets you deal with each part separately. Next, evaluate \(2^3\).
  • \(2^3 = 2 \times 2 \times 2 = 8\)
Then, the expression simplifies to \(8x^3\). Simplifying expressions is all about making the pieces easy to manage and understand, turning potentially complex math into something straightforward.

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 a graphing utility to determine whether the divisions have been performed correctly. Graph each side of the given equation in the same viewing rectangle. The graphs should coincide. If they do not, correct the expression on the right side by using polynomial division. Then use your graphing utility to show that the division has been performed correctly. $$\frac{x^{3}+3 x^{2}+5 x+3}{x+1}=x^{2}-2 x+3$$

You just signed a contract for a new job. The salary for the first year is \(\$ 30,000\) and there is to be a percent increase in your salary each year. The algebraic expression $$\frac{30,000 x^{n}-30,000}{x-1}$$ describes your total salary over n years, where \(x\) is the sum of 1 and the yearly percent increase, expressed as a decimal. a. Use the given expression and write a quotient of polynomials that describes your total salary over three years. b. Simplify the expression in part (a) by performing the division. c. Suppose you are to receive an increase of \(5 \%\) per year. Thus, \(x\) is the sum of 1 and \(0.05,\) or \(1.05 .\) Substitute 1.05 for \(x\) in the expression in part (a) as well as in the simplified form of the expression in part (b). Evaluate each expression. What is your total salary over the three-year period?

$$\text { Graph: } y=\frac{1}{3} x$$

Use a graphing utility to graph each side of the equation in the same viewing rectangle. (Call the left side \(y_{1}\) and the right side \(y_{2} .\) I If the graphs coincide, verify that the multiplication has been performed correctly. If the graphs do not appear to coincide, this indicates that the multiplication is incorrect. In these exercises, correct the right side of the equation. Then graph the left side and the corrected right side to verify that the graphs coincide. \((x-2)(x+2)+4=x^{2} ;\) Use a \([-6,5,1]\) by \([-2,18,1]\) viewing rectangle.

Solve: \(0.02(x-5)=0.03-0.03(x+7) .\) (Section 2.3 Example 5 )
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied 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.