Problem 66 Simplify \(\sqrt[5]{128}\)... [FREE SOLUTION]

Chapter 10: Problem 66

Simplify \(\sqrt[5]{128}\)

Short Answer

Expert verified

2^{1.4}

Step by step solution

Prime Factorization of 128

First, find the prime factorization of 128. Start by dividing by the smallest prime number, which is 2, and continue until the quotient is 1. The factorization is: 128 梅 2 = 64 64 梅 2 = 32 32 梅 2 = 16 16 梅 2 = 8 8 梅 2 = 4 4 梅 2 = 2 2 梅 2 = 1 So, 128 = 2^7.

Express under the Radical Sign

Rewrite the expression \(\sqrt[5]{128}\) using the prime factorization from step 1: \[\sqrt[5]{128} \rightarrow \sqrt[5]{2^7}\].

Apply Radical Rules

Use the property \(\sqrt[n]{a^m} = a^{\frac{m}{n}}\) to simplify the expression: \(\sqrt[5]{2^7} = 2^{\frac{7}{5}}\).

Simplify the Exponent

Determine the numerical value of the exponent, which is \(2^{\frac{7}{5}}\), and leave it in its simplified radical form. Given that fractions denote roots, \(2^{1.4} \approx 3.03 \: Less precise \sqrt[5]{2^7} = 2^{1.4}\).

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.

Prime Factorization

Prime factorization is a method used to express a number as the product of its prime numbers. Prime numbers are numbers that are only divisible by 1 and themselves. Here is how you factorize 128:

Start by dividing by the smallest prime number, which is 2.
Keep dividing by 2 until the quotient is 1.

So it looks like this:
128 梅 2 = 64
64 梅 2 = 32
32 梅 2 = 16
16 梅 2 = 8
8 梅 2 = 4
4 梅 2 = 2
2 梅 2 = 1
Thus, the prime factorization of 128 is: \(128 = 2^7\).
This breakdown helps to simplify radical expressions.

Radical Expressions

A radical expression involves roots such as square roots, cube roots, etc.
The symbol for radicals is \(\text{鈭殅\) or \(\sqrt[n]{}\), where n indicates the degree of the root.
To simplify a radical, use prime factorization.
Let's revisit the problem \(\sqrt[5]{128}\).
Using the prime factorization \(128 = 2^7\), the expression under the radical becomes: \(\sqrt[5]{128} = \sqrt[5]{2^7}\).
This form helps us apply the property of fractional exponents for further simplification.

Fractional Exponents

Fractional exponents express roots as a power. The general rule is \(\sqrt[n]{a^m} = a^{\frac{m}{n}}\).
For the exercise: \(\sqrt[5]{128}\), we have:
\(\sqrt[5]{2^7} = 2^{\frac{7}{5}}\).
This transformation allows calculation using exponent rules.
Fractional exponents might seem tricky, but they鈥檙e just another way to represent roots:

\(2^{\frac{1}{2}} = \sqrt{2}\)
\(2^{\frac{1}{3}} = \sqrt[3]{2}\)
\(2^{\frac{2}{3}} = \sqrt[3]{2^2}\)

For our case, leave it as \(2^{1.4}\), approximated numerically as around 3.03. Understanding this helps simplify and compute roots efficiently.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Simplify \(\sqrt[5]{128}\)

Short Answer

Step by step solution

Prime Factorization of 128

Express under the Radical Sign

Apply Radical Rules

Simplify the Exponent

Key Concepts

Prime Factorization

Radical Expressions

Fractional Exponents

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Logic and Functions

Mechanics Maths

Theoretical and Mathematical Physics

Probability and Statistics

Statistics

Study anywhere. Anytime. Across all devices.