/*! 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 31 Simplify the given expressions. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 31

Simplify the given expressions. Express results with positive exponents only. $$\begin{aligned} &2 v^{4}\\\ &(2 v)^{4} \end{aligned}$$

Short Answer

Expert verified
The expressions simplified are \(2v^4\) and \(16v^4\).

Step by step solution

01

Analyze the Expression \(2v^4\)

In the expression \(2v^4\), the coefficient is 2, and \(v^4\) represents \(v\) raised to the fourth power. This expression is already in its simplest form since there are no parentheses affecting it, and the power of \(v\) is positive.
02

Simplify the Expression \((2v)^4\)

To simplify \((2v)^4\), use the property of exponents that states \((ab)^n = a^n b^n\). Here, \((2v)^4 = 2^4 \cdot v^4\). First, compute \(2^4 = 16\), so the expression simplifies to \(16v^4\).
03

Combine Results with Positive Exponents

Now that both parts of the expression are simplified, write them together. The overall expression becomes: \(2v^4\) and \(16v^4\), which are both expressed with positive exponents.

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.

Exponents
Exponents are a fundamental part of algebra and mathematics in general. When you see an exponent, it indicates how many times you multiply a number by itself. For example, in the expression \(v^4\), the base \(v\) is multiplied by itself four times: \(v \times v \times v \times v\). This is a powerful tool that allows us to express large numbers compactly and to perform calculations more easily.
  • **Positive exponents**: These indicate normal multiplication. The above example \(v^4\) is a positive exponent since it involves multiplying \(v\) by itself four times.
  • **Negative exponents**: Although not part of this specific exercise, negative exponents represent division or inverse multiplication, such as \(v^{-4} = \frac{1}{v^4}\).
Understanding how to work with exponents allows you to manipulate and simplify complex expressions effectively. This is essential for performing operations like simplification.
Simplification
Simplification in algebra means reducing an expression to its simplest form. This involves combining like terms, performing arithmetic operations, and using algebraic laws like those of exponents.
  • **Breaking down expressions**: Consider the expression \((2v)^4\). By breaking it into parts, it becomes easier to manage. Use the property \((ab)^n = a^n \times b^n\), which simplifies our expression to \(2^4 \times v^4\).
  • **Calculate powers**: Compute \(2^4\), which is \(16\). Therefore the expression \((2v)^4\) simplifies to \(16v^4\), a much simpler format.
Simplification helps us see expressions more clearly and makes further arithmetic or algebraic operations straightforward and manageable.
Mathematical Expressions
Mathematical expressions are combinations of numbers, variables, operations, and sometimes exponents. They are the building blocks of algebra and help us represent real-world problems mathematically so we can solve them systematically.
  • **Components of expressions**: In the exercise, \(2v^4\) and \((2v)^4\) are our expressions. Each contains numbers (coefficients like 2), variables (\(v\)), and operations (exponents).
  • **Importance of structure**: The way expressions are structured affects how they are simplified and solved. Understanding this structure is crucial for carrying out correct operations.
  • **Grouping and ordering**: Parentheses in expressions like \((2v)^4\) tell us to handle operations within them first, indicating priority. Removing these can be part of simplifying the expression.
Recognizing and manipulating mathematical expressions accurately is a key skill in algebra that aids in problem-solving and simplifying tasks.

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

Perform the indicated operations. The first number is approximate, and the second number is exact. $$8.62 \div 1728$$

Solve the given problems. Refer to Appendix B for units of measurement and their symbols. A flare is shot up from the top of a tower. Distances above the flare gun are positive and those below it are negative. After 5 s the vertical distance (in \(\mathrm{ft}\) ) of the flare from the flare gun is found by evaluating \((70)(5)+(-16)(25) .\) Find this distance.

Solve the given problems. Refer to Appendix B for units of measurement and their symbols. A baseball player's batting average (total number of hits divided by total number of at-bats) is expressed in decimal form from 0.000 (no hits for all at-bats) to 1.000 (one hit for each at-bat). A player's batting average is often shown as 0.000 before the first at-bat of the season. Is this a correct batting average? Explain.

Perform the indicated operations assuming all numbers are approximate. Round your answers using the procedure shown in Example 11 $$4.52-\frac{2.056(309.6)}{395.2}$$

Solve the given problems. A tsunami is a very high ocean tidal wave (or series of waves) often caused by an earthquake. An Alaskan tsunami in 1958 measured over \(500 \mathrm{m}\) high; an Asian tsunami in 2004 killed over 230,000 people; a tsunami in Japan in 2011 killed over 10,000 people. An equation that approximates the speed \(v\) (in \(\mathrm{m} / \mathrm{s}\) ) of a tsunami is \(\left.v=\sqrt{g d}, \text { where } g=9.8 \mathrm{m} / \mathrm{s}^{2} \text { and } d \text { is the average depth (in } \mathrm{m}\right)\) of the ocean floor. Find \(v\) (in \(\mathrm{km} / \mathrm{h}\) ) for \(d=3500 \mathrm{m}\) (valid for many parts of the Indian Ocean and Pacific Ocean).
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

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.