/*! 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 36 Rational Exponents Write an eq... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 36

Rational Exponents Write an equivalent expression using exponential notation. $$\sqrt[4]{10}$$

Short Answer

Expert verified
The expression \(\textstyle \sqrt[4]{10}\) is equivalent to \(\textstyle 10^{\frac{1}{4}}\).

Step by step solution

01

Understand the Radical Notation

The given expression is written in radical notation, which is \(\textstyle \sqrt[4]{10}\). This represents the fourth root of 10.
02

Rewrite the Radical as an Exponent

In exponential notation, the n-th root of a number can be written as that number raised to the power of \(\frac{1}{n}\). For this specific case, \(\textstyle \sqrt[4]{10}\) can be rewritten as \(\textstyle 10^{\frac{1}{4}}\).

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.

radical notation
Radical notation is the way to express roots of numbers. When you see a symbol that looks like \( \sqrt{} \), it means you're dealing with a root. For example, \( \sqrt{4} \) represents the square root of 4. This can be read as 'the number which, when multiplied by itself, gives the inside number.'

It's possible to work with various roots using a small number inside the radical sign. For instance, \( \sqrt[3]{8} \) stands for the cube root of 8. If no number is shown, it means a square root by default.
  • The general form of radical notation is \( \sqrt[n]{a} \)
  • Here, 'n' is the root's degree, and 'a' is the number you're finding the root of
  • So, \( \sqrt[n]{a} \) means the number which, when raised to 'n', gives 'a'

This format is very handy but not always the best for complex calculations. That's where exponential notation can be a better option.
exponential notation
Exponential notation is another way to represent roots, especially useful for simplifying and solving equations. It uses exponents, which show how many times a number is to be multiplied by itself.

In this approach, the \( n \)-th root of a number can be rewritten using a fraction as the exponent. For example, \( \sqrt[4]{10} \) can be expressed as \( 10^{\frac{1}{4}} \). Here鈥檚 how:
  • The denominator of the fraction (4 in this case) tells you the degree of the root
  • The numerator (1 here) is the power to which the number is raised

This form helps to transition between different mathematical operations easily, especially when combining or comparing different expressions. Converting radicals to exponentials makes manipulating them in algebra much more straightforward.
n-th root
The n-th root is a broader concept of finding roots beyond the common square or cube roots. It generalizes the idea to any degree. So, when we say 'n-th root', the 'n' specifies what root you're finding.

The concept can be written in either form:
  • Radical Notation: \( \sqrt[n]{a} \)
  • Exponential Notation: \( a^{\frac{1}{n}} \)

These two forms are interchangeable, and knowing both makes you flexible in solving mathematical problems. For \( \sqrt[4]{10} \), it translates to \( 10^{\frac{1}{4}} \) which means the number that, when raised to the power of 4, results in 10.

This understanding is crucial for working efficiently with higher-degree roots and for simplifying more complex algebraic expressions.

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. $$5 i^{5}+4 i^{3}$$

Simplify. $$(5 i)^{3}$$

Find the midpoint of the segment with the given endpoints. $$ (-1,2) \text { and }(1,-3) $$

Simplify. $$i^{2}+i^{4}$$

The absolute value of a complex number \(a+b i\) is its distance from the origin. (See the graph above.) Using the distance formula, we have \(|a+b i|=\sqrt{a^{2}+b^{2}}\) Find the absolute value of each complex number. $$|3+4 i|$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Geometry

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.