/*! 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 83 Simplify. Assume that all variab... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 83

Simplify. Assume that all variables represent positive real numbers. \(\sqrt[4]{\frac{1}{16} r^{8} t^{20}}\)

Short Answer

Expert verified
\( \frac{1}{2} r^2 t^5 \)

Step by step solution

01

- Rewrite the expression with a common base

Rewrite the fraction inside the radical using the property of exponents: \[ 16 = 2^4 \] Hence, \[ \frac{1}{16} = \frac{1}{2^4} = 2^{-4} \] Now, the expression becomes: \[ \frac{1}{16} r^8 t^{20} = 2^{-4} r^8 t^{20} \]
02

- Apply the fourth root to each term separately

Use the property of radicals \[ \frac{\big( ab \big)^n} = \big( a^n \big) \big( b^n \big) \] to simplify: \[ \big( 2^{-4} r^8 t^{20} \big)^{\frac{1}{4}} \]
03

- Simplify each term inside the radical

Apply \[ \big( a^m \big)^n = a^{m \frac{n}} \] to each term: \[ 2^{-4\frac{1}{4}} = 2^{-1} \], \[ r^{8\frac{1}{4}} = r^2 \], and \[ t^{20\frac{1}{4}} = t^5 \]
04

- Rewrite the simplified terms

Combine the simplified terms to get: \[ 2^{-1} r^2 t^5 \]
05

- Simplify the expression

Since \[ 2^{-1} = \frac{1}{2} \], rewrite the expression as: \[ \frac{1}{2} r^2 t^5 \]

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 tell us how many times a number, the base, is multiplied by itself. For example, in the expression \(2^4\), the base is 2, and it is multiplied by itself 4 times to get 16. When simplifying expressions with exponents, remember these key rules:

  • \(a^m \times a^n = a^{m+n}\)
  • \(\frac{a^m}{a^n} = a^{m-n}\)
  • \((a^m)^n = a^{m \times n}\)
Using these properties, you can rewrite and simplify expressions more easily.
properties of radicals
Radicals are expressions that involve roots, such as square roots \(\sqrt{}\) or fourth roots \(\sqrt[4]{}\). There are some important properties of radicals that help in simplification:

  • \(\sqrt[n]{a} \times \sqrt[n]{b} = \sqrt[n]{a \times b}\)
  • \(\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}\)
  • \((a^m)^{1/n} = a^{m/n}\)
These properties allow us to break down complex radical expressions into simpler parts. For example, in the exercise, rewriting \(\frac{1}{16}\) as \(2^{-4}\) helped make the radical expression easier to handle.
simplifying radicals
Simplifying radicals involves reducing the expression inside the radical to its simplest form. Here鈥檚 a step-by-step guide on how to simplify radicals:

  1. Factorize the expression inside the radical.
  2. Use the properties of radicals to separate the terms.
  3. Simplify each term individually using the property \((a^m)^n = a^{m \times n}\).
In the exercise, we simplified \(\sqrt[4]{2^{-4} r^8 t^{20}}\) by breaking it down into three separate terms and then simplifying each using the property of exponents.
fractional exponents
Fractional exponents provide an alternate way to represent roots. For example, the fourth root of \(a\) can be written as \(a^{1/4}\). This can be very helpful in simplification. Some important points to remember about fractional exponents are:

  • \((a^m)^{1/n} = a^{m/n}\), so \(\sqrt[n]{a^m} = a^{m/n}\)
  • \(a^{m/n} = \sqrt[n]{a^m}\)
  • Simplifying a term like \(a^{8/4}\) involves just dividing the exponent, resulting in \(a^2\).
Using these properties, we can simplify expressions like \(\sqrt[4]{r^8}\) to \(r^2\) and \(\sqrt[4]{t^{20}}\) to \(t^5\). This method was used step-by-step in the exercise solution to break down the expression.

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

Find the distance between each pair of points. \((\sqrt{7}, 9 \sqrt{3})\) and \((-\sqrt{7}, 4 \sqrt{3})\)

Simplify. Assume that all variables represent positive real numbers. \(\sqrt[3]{5} \cdot \sqrt{6}\)

Simplify. Assume that all variables represent positive real numbers. \(\sqrt[3]{-16 z^{5} t^{7}}\)

List all of the following sets to which each number belongs. A number may belong to more than one set. real numbers pure imaginary numbers nonreal complex numbers complex numbers $$ \frac{13}{3} $$

Write each radical as an exponential and simplify. Leave answers in exponential form. Assume that all variables represent positive numbers. $$ \sqrt[3]{\sqrt{k}} $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

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.