/*! 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 102 Simplify by reducing the index o... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 102

Simplify by reducing the index of the radical. $$ \sqrt[4]{7^{2}} $$

Short Answer

Expert verified
The simplified form of \(\sqrt[4]{7^{2}}\) is \(\sqrt{7}\).

Step by step solution

01

Translating the Radical

Transform the fourth root of a square into an exponent using the property \(\sqrt[n]{a^{m}} = a^{m/n}\). This translates \(\sqrt[4]{7^{2}}\) into the form \(7^{2/4}\).
02

Simplify the Exponents

Further simplify the exponents by dividing 2 by 4 to get 0.5. Hence, the final form of the equation becomes \(7^{0.5}\).
03

Final Simplification

Express \(7^{0.5}\) in radical form since that is more simplified in this case. Thus \(7^{0.5}\) becomes \(\sqrt{7}\).

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
When working with rational exponents, understanding the conversion between radicals and exponents is key. Rational exponents are those in the form of fractions, where the numerator represents the power and the denominator indicates the root. For example, the expression \(a^{m/n}\) indicates that the base \(a\) is raised to the power \(m\) and then the \(n\)-th root is taken.
  • The expression \(a^{1/n}\) is equivalent to the \(n\)-th root of \(a\).
  • \(a^{m/n}\) is equivalent to \(\sqrt[n]{a^m}\).
This is why in the original exercise, \(\sqrt[4]{7^2}\) can be rewritten as \(7^{2/4}\). This interchangeability helps when simplifying expressions and solving equations.
Radical Expressions
Radical expressions involve roots, such as square roots or cube roots, and are often rewritten using rational exponents for easier manipulation. The index of a radical specifies the degree of the root. For example, in \(\sqrt[4]{7^2}\), 4 is the index showing a fourth root is required.
  • Redefining radicals into rational exponents makes it simpler to apply exponentiation rules.
  • Use rational exponents to simplify difficult radical expressions to uncover more straightforward forms.
In this exercise, converting \(\sqrt[4]{7^2}\) to \(7^{2/4}\) allows for simplifying the fraction \(\frac{2}{4}\) to get \(7^{0.5}\). Ultimately, returning to radical form, \(7^{0.5}\) equals \(\sqrt{7}\), a simple radical expression.
Exponentiation Rules
Exponentiation rules facilitate simplifying expressions with exponents. These rules are particularly beneficial when transitioning between radical and exponential forms. Key exponentiation rules include:
  • Product Rule: \(a^m \cdot a^n = a^{m+n}\)
  • Quotient Rule: \(\frac{a^m}{a^n} = a^{m-n}\)
  • Power Rule: \((a^m)^n = a^{m \times n}\)
  • Rational Exponent Rule: \(a^{m/n} = \sqrt[n]{a^m}\)
For the problem at hand, the Rational Exponent Rule was vital in transforming the fourth root \(\sqrt[4]{7^2}\) to \(7^{2/4}\), allowing us to manipulate it more straightforwardly. Understanding these rules and how they interconnect helps in easing complex problems into simpler, more manageable parts.

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

Simplify by reducing the index of the radical. $$ \sqrt[9]{x^{6}} $$

Factor Completely. $$(y+1)^{3}+1$$

Determine whether each statement is trueor false. If the statement is false, make the necessary change(s) toproduce a true statement. $$x^{2}+36=(x+6)^{2}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Some rational numbers are not positive.

Explain how to add \(\sqrt{3}+\sqrt{12}\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Pure 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.