/*! 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 105 Write each expression in radical... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 105

Write each expression in radical form. $$ x^{\frac{p}{n}} $$

Short Answer

Expert verified
\(x^\frac{p}{n}\) is equivalent to the radical form \(\sqrt[n]{x^p}\).

Step by step solution

01

Identification of the base, exponent's numerator and denominator

We identify three components: \(x\) which is our base, \(p\) or the numerator of the exponent, and \(n\) or the denominator of the exponent.
02

Conversion to radical form

We apply the rule that an expression \(x^\frac{p}{n}\) is equivalent to the nth root of \(x^p\). So, we have \(\sqrt[n]{x^p}\)
03

Simplification

This is already in simplest form, so no further simplification is necessary.

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.

Rational Exponents
Rational exponents might seem intimidating at first, but they are just a different way to represent roots and powers. To break it down, a rational exponent looks like this: \( x^{\frac{p}{n}} \). Here, the numerator \( p \) stands for the power, while the denominator \( n \) indicates the root.
There are several benefits to using rational exponents:
  • They simplify the process of combining roots and exponents in mathematical operations.
  • They provide a more flexible framework for writing and manipulating expressions.
By understanding and practicing with rational exponents, you'll be able to handle complex expressions with greater ease and clarity.
Radical Form
The radical form of an expression is a way to write exponents as roots. When you see a radical sign \( \sqrt{} \), it represents the root of a number. In the case of \( x^{\frac{p}{n}} \), converting it to radical form means finding the \( n \)th root of \( x^p \). This is written as \( \sqrt[n]{x^p} \).
Understanding how to convert between radical form and rational exponents is useful because it offers two different perspectives:
  • Radical form can make certain types of calculations, especially those involving roots, more intuitive.
  • It also allows for easier comparison and simplification in problems dealing with roots and exponents.
Mastering radical form makes you versatile in tackling problems involving both roots and powers.
Nth Roots
Nth roots are a generalization of square roots. While a square root specifically involves "2," the nth root allows for any number \( n \). Essentially, when you take the nth root of a number, you're asking "what number, multiplied by itself \( n \) times, will give me my original number?" For example, the 3rd root (or cube root) asks what number, cubed, yields the original number.
Nth roots have applications in various areas:
  • They are used in algebra for simplifying expressions and solving equations.
  • They appear in scientific calculations, such as determining growth rates over time.
By understanding nth roots, you can interpret and solve a wider range of mathematical and real-world problems. This knowledge is crucial for anyone working in fields that rely on mathematics.

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

Which expression is equal to \(\log _{5} x+4 \cdot \log _{5} y-2 \cdot \log _{5} z ?\) \(\begin{array}{llll}{\text { A. } \log _{5}(-8 x y z)} & {\text { B. }-\log _{5} \frac{4 x y}{2 z}} & {\text { C. } \log _{5} \frac{(x y)^{4}}{z^{2}}} & {\text { D. } \log _{5} \frac{x y^{4}}{z^{2}}}\end{array}\)

Solve each equation. Check your answers. $$ \log 6 x-3=-4 $$

Solve each equation. If necessary, round to the nearest ten-thousandth. $$ 4 \log _{3} 2-2 \log _{3} x=1 $$

Acoustics In Exercises \(76-78\) , the loudness measured in decibels \((d B)\) is defined by loudness \(=10\) log \(\frac{I}{I_{0}},\) where \(I\) is the intensity and \(I_{0}=10^{-12} \mathrm{W} / \mathrm{m}^{2}\) . A screaming child can reach 90 dB. A launch of the space shuttle produces sound of 180 \(\mathrm{dB}\) at the launch pad. a. Find the intensity of each sound. b. How many times as intense as the noise from a screaming child is the noise from a shuttle launch?

Write each logarithm as the quotient of two common logarithms. Do not simplify the quotient. $$ \log _{4} 3 x $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

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.