/*! 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 20 Rewrite each expression so that ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 20

Rewrite each expression so that no fraction appears in the exponent and each expression is in the form \(a^{x}\). a. \(3^{w / 4}\) b. \(2^{x / 3}\) c. \(\left(\frac{1}{2}\right)^{x_{4}}\) d. \(\left(\frac{1}{4}\right)^{x / 2}\)

Short Answer

Expert verified
a. \( (3^{1/4})^w \) b. \( (2^{1/3})^x \) c. \( 2^{-x_{4}} \) d. \( 4^{-x/2} \)

Step by step solution

01

Understand the Problem

The goal is to rewrite each expression so that no fractions appear in the exponents. Each expression should be in the form of a power of some base.
02

Use Property of Exponents

Recall that fractions in exponents can be rewritten using the property \(a^{m/n} = (a^{1/n})^m\). This will help in converting all expressions into the form \(a^x\).
03

Rewrite Expression (a)

Given expression: \(3^{w / 4}\). Using the property of exponents, \(3^{w / 4} = (3^{1 / 4})^w\). Thus, the expression is \(a^x\) form where \(a = 3^{1 / 4}\) and \(x = w\).
04

Rewrite Expression (b)

Given expression: \(2^{x / 3}\). Using the property of exponents, \(2^{x / 3} = (2^{1 / 3})^x\). Thus, the expression is in \(a^x\) form where \(a = 2^{1 / 3}\) and \(x = x\).
05

Rewrite Expression (c)

Given expression: \left(\frac{1}{2}\right)^{x_{4}}\. Using the property of exponents, \left(\frac{1}{2}\right)^{x_{4}} = \left(2^{-1}\right)^{x_{4}} = 2^{-x_{4}}\. Thus, the expression is in \(a^x\) form where \(a = 2\) and \(x = -x_{4}\).
06

Rewrite Expression (d)

Given expression: \left(\frac{1}{4}\right)^{x / 2}\. Using the property of exponents, \left(\frac{1}{4}\right)^{x / 2} = \left(4^{-1}\right)^{x / 2} = 4^{-x/2}\. Thus, the expression is in \(a^x\) form where \(a = 4^{-1} = \left(2^2\right)^{-1} = 2^{-2}\) and \(x = x/2\).

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.

Properties of Exponents
Understanding the properties of exponents is essential when dealing with algebraic expressions that involve powers. One key property is the rule for fractional exponents. This can be expressed as \(a^{m/n} = (a^{1/n})^m\). This means that any exponent with a fraction can be transformed into a different but equivalent exponentiation form that helps simplify or reformat the expression.
Algebraic Expressions
Algebraic expressions can be understood as a combination of numbers, variables, and operations. These operations can include addition, subtraction, multiplication, division, and exponentiation. Expressions with exponents often appear in algebra, and transforming these expressions to a simpler form can help in solving problems or understanding the relationships between variables. For example, expressions like \(3^{w/4}\) or \(2^{x/3}\) can be rewritten using the rules of exponents, making them more workable and easier to understand.
Fractional Exponents
Fractional exponents represent both a root and a power simultaneously. For instance, in the exercise, expressions like \(3^{w/4}\) and \(2^{x/3}\) feature fractions in their exponents. By using the property \(a^{m/n} = (a^{1/n})^m\), these fractional exponents can be rewritten. This transforms the expression from having fractional exponents to a form where the base is a root and the exponent is an integer. For instance: \(3^{w/4} = (3^{1/4})^w\) shows how the fractional exponent becomes an integer exponent, simplifying computations 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

a. Complete the following table for the exponential function \(y=20(0.75)^{x}\) b. Choose the correct word in each italicized pair to describe the function: For the exponential function \(y=20(0.75)^{x}\), the differences are constant/decreasing in magnitude and the ratios are constant/decreasing in magnitude.

A swimming pool is initially shocked with chlorine to bring the chlorine concentration to 3 ppm (parts per million). Chlorine dissipates in reaction to bacteria and sun at a rate of about \(15 \%\) per day. Above a chlorine concentration of 2 ppm, swimmers experience burning eyes, and below a concentration of 1 ppm, bacteria and algae start to proliferate in the pool environment. a. Construct an exponential decay function that describes the chlorine concentration (in parts per million) over time. b. Construct a table of values that corresponds to monitoring chlorine concentration for at least a 2-week period. c. How many days will it take for the chlorine to reach a level tolerable for swimmers? How many days before bacteria and algae will start to grow and you will need to add more chlorine? Justify your answers.

Given an initial value of 50 units for parts (a)-(d) below, in each case construct a function that represents \(Q\) as a function of time \(t\). Assume that when \(t\) increases by 1: a. \(Q(t)\) doubles b. \(Q(t)\) increases by \(5 \%\) c. \(Q(t)\) increases by ten units d. \(Q(t)\) is multiplied by 2.5

An ancient king of Persia was said to have been so grateful to one of his subjects that he allowed the subject to select his own reward. The clever subject asked for a grain of rice on the first square of a chessboard, two grains on the second square, four on the next, and so on. a. Construct a function that describes the number of grains of rice, \(G,\) as a function of the square, \(n,\) on the chessboard. (Note: There are 64 squares.) b. Construct a table recording the numbers of grains of rice on the first ten squares. c. Sketch your function. d. How many grains of rice would the king have to provide for the 64 th (and last) square?

(Graphing program recommended.) A small village has an initial size of 50 people at time \(t=0,\) with \(t\) in years. a. If the population increases by 5 people per year, find the formula for the population size \(P(t)\). b. If the population increases by a factor of 1.05 per year, find a new formula \(Q(t)\) for the population size. c. Plot both functions on the same graph over a 30 -year period. d. Estimate the coordinates of the point(s) where the graphs intersect. Interpret the meaning of the intersection point(s).
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

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.