/*! 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 81 Simplify completely. Assume the ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 81

Simplify completely. Assume the variables represent positive real numbers. The answer should contain only positive exponents. \(\left(32 r^{1 / 3} s^{4 / 9}\right)^{3 / 5}\)

Short Answer

Expert verified
The simplified expression is: \(8r^{\frac{1}{5}}s^{\frac{4}{15}}\).

Step by step solution

01

Distribute the exponent

In this step, we will distribute the exponent \(3/5\) to the terms inside the parentheses: \(\left(32 r^{1 / 3} s^{4 / 9}\right)^{3 / 5} = 32^{3/5} \cdot r^{\frac{1}{3} \cdot \frac{3}{5}} \cdot s^{\frac{4}{9} \cdot \frac{3}{5}}\)
02

Simplify the exponents

Now we will simplify the exponents by performing the multiplications: \(32^{3/5} \cdot r^{\frac{3}{15}} \cdot s^{\frac{12}{45}}\)
03

Reduce the fractions

Reduce the fractions in the exponents to the simplest form: \(32^{3/5} \cdot r^{\frac{1}{5}} \cdot s^{\frac{4}{15}}\)
04

Simplify the power term

Simplify the power term \(32^{3/5}\) by finding the prime factorization of the base (32) and then raising the prime factors to the given power: \(32 = 2^5\) Now raise \(2^5\) to the power of \(3/5\): \((2^5)^{3/5} = 2^{\left(5 \cdot \frac{3}{5}\right)} = 2^3\)
05

Rewrite the expression

Now we rewrite the simplified expression with the simplified power term: \(2^3 \cdot r^{\frac{1}{5}} \cdot s^{\frac{4}{15}}\)
06

Final expression

The completely simplified expression is: \(8r^{\frac{1}{5}}s^{\frac{4}{15}}\)

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, often called powers, indicate how many times a number, known as the base, is multiplied by itself. For example, in the expression \(a^n\), \(a\) is the base and \(n\) is the exponent. This means the base \(a\) is multiplied by itself \(n\) times.
  • Exponents are a shorthand way to express repeated multiplication.
  • For instance, \(3^4\) means \(3 \times 3 \times 3 \times 3 = 81\).
  • Exponents can be used with any real numbers as the base.
  • Fractional exponents, in contrast, indicate roots or combinations of roots and powers.
Understanding how to manipulate exponents by applying multiplication or division rules can simplify complex algebraic expressions significantly. This fundamental knowledge serves as a building block in mathematics, allowing us to work efficiently with powers in various scenarios.
Fractional Exponents
Fractional exponents, such as \(a^{m/n}\), describe a root and a power within the same expression. The numerator \(m\) indicates the power to which the base is raised, while the denominator \(n\) signifies the root.
  • \(a^{1/n}\) is equivalent to the \(n\)-th root of \(a\).
  • For example, \(a^{1/2}\) is the square root of \(a\), denoted \(\sqrt{a}\).
  • Similarly, \(a^{m/n}\) is equal to \((\sqrt[n]{a})^m\).
Using fractional exponents helps in simplifying radical expressions and performing root operations more systematically. In our exercise, converting expressions like \(32^{3/5}\) makes it manageable to calculate, as it combines root taking with power raising. It effectively transforms radical expressions into more straightforward terms to handle mathematically.
Prime Factorization
Prime factorization involves breaking down a composite number into smaller prime numbers multiplied together. This method is particularly helpful in simplifying expressions involving powers.
  • A prime number is a number greater than 1 that has no divisors other than 1 and itself. Examples include 2, 3, 5, and 7.
  • Prime factorization expresses a number as a product of its prime factors.
  • For example, the prime factorization of 32 is \(2^5\), indicating it consists of multiplying five 2's together.
In mathematics, prime factorization is often used to simplify powers, especially when dealing with fractional exponents. The process allows us to decompose numbers like 32 into a base of prime numbers, which is particularly useful when we need to apply exponents fully, as seen in the term \((2^5)^{3/5}\). Understanding how to find and use the prime factorization of a number ensures accurate and simplified expression manipulation.

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 completely. $$\frac{-10-\sqrt{50}}{5}$$

Perform the indicated operation and simplify. Assume all variables represent positive real numbers. $$\frac{\sqrt{21 z^{18}}}{\sqrt{3 z^{13}}}$$

Rationalize the denominator of each expression. $$\frac{25}{\sqrt{10}}$$

Rationalize the denominator and simplify completely. Assume the variables represent positive real numbers. $$\frac{3}{2+\sqrt{3}}$$

Rationalize the denominator of each expression. Assume all variables represent positive real numbers. $$\sqrt[5]{\frac{3}{8}}$$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

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.