/*! 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 29 Rational Exponents Write an eq... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 29

Rational Exponents Write an equivalent expression using radical notation and, if possible, simplify. $$125^{4 / 3}$$

Short Answer

Expert verified
625

Step by step solution

01

- Convert to Radical Notation

To convert a rational exponent into radical notation, rewrite the expression using the property \(a^{m/n} = \sqrt[n]{a^{m}}\). In this case, rewrite \(125^{4/3}\) as \((125^{4})^{1/3}\) which is \sqrt[3]{125^4}\.
02

- Simplify the Radical Expression

Recognize that \(125 = 5^3\). Substituting this into our expression, we have \(\root[3]{(5^3)^4}\). Simplify the exponent inside the radical: \( (5^3)^4 = 5^{12} \).
03

- Simplify Further Under the Radical

Applying the property of exponents, we get \(\root[3]{5^{12}} = 5^{12/3} = 5^4\). Calculate \(5^4\) to find the final simplified result.
04

- Calculate the Power

Finally, compute \(5^4 = 5 \times 5 \times 5 \times 5 = 625\). Therefore, the radical expression for \(125^{4/3}\) is simplified to 625.

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 a way to represent the root of a number. For instance, the square root of a number is written using a radical symbol (鈭�). When you see \(\root[n]{a}\), this means the n-th root of a. Radical notation is especially useful when working with rational exponents. When converting rational exponents to radical notation, use the property: \({a^{m/n} = \root[n]{a^m}}\). This means you take the n-th root of the base raised to the m-th power. For example, converting \({125^{4/3}}\) to radical notation gives us \(\root[3]{125^4}\). This means we're taking the cube root of 125 raised to the 4th power.
Exponent Properties
Understanding exponent properties makes complex expressions a lot easier to handle. Some of the main properties you should be familiar with include:
  • Product of Powers: When multiplying like bases, add the exponents. \({a^m \times a^n = a^{m+n}}\).
  • Power of a Power: When raising a power to another power, multiply the exponents. \({(a^m)^n = a^{m \times n}}\).
  • Power of a Product: When raising a product to a power, apply the exponent to each factor. \({(ab)^n = a^n \times b^n}}\).
Using these properties, you can simplify complex algebraic expressions. In the exercise, we used the property \({(a^m)^n = a^{m \times n}}\) to simplify \(\root[3]{(5^3)^4}\) to \(\root[3]{5^{12}}\).
Simplifying Exponents
Simplifying exponents often involves breaking down the expressions using exponent properties. Take a step-by-step approach:
  • Identify the base and the exponents.
  • Apply known properties to simplify.
  • Convert between different forms like from radical to rational notation.
In our example, we started by recognizing that 125 equals \({5^3}\). Rewriting \({125^{4/3}}\) as \({(5^3)^{4/3}}\) allowed us to simplify the expression using the property \({(a^m)^n = a^{m \times n}}\). This resulted in \({5^{12/3}}\), which simplifies to \({5^4}\). Finally, calculating \({5^4}\) gave us the result 625. Simplification is all about breaking the problem into smaller, manageable steps.
Radicals
Radicals involve roots of numbers and are denoted using the radical symbol. Here are some key points:
  • Square Roots: Often written as \(\sqrt{a}\), mean \(\root[2]{a}\).
  • Cubic Roots: Written as \(\root[3]{a}\), meaning the number which, when cubed, gives 'a'.
  • Higher-order Roots: Generalized as \(\root[n]{a}\).
Radicals are used to 'undo' exponents. For example, \(\root[3]{8} = 2\), because \({2^3 = 8}\). In the exercise, converting \({125^{4/3}}\) into \(\root[3]{125^4}\) involved recognizing the radical as a way to simplify exponential expressions. Radicals help us understand the depth of numbers and their roots.

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. Where appropriate, find an approximation to three decimal places. $$ (0.5,100) \text { and }(1.5,-100) $$

A cube measures \(5 \mathrm{cm}\) on each side. How long is the diagonal that connects two opposite corners of the cube? Give an exact answer. (Image can't copy)

Simplify. $$(-i)^{4}$$

Simplify. $$(-i)^{6}$$

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. Write the answer using radical notation. $$ \sqrt[3]{a} \sqrt[6]{a} $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

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.