/*! 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 123 Simplify each expression. Assume... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 123

Simplify each expression. Assume that variables represent positive integers. $$ \left(x^{-3 s}\right)^{3} $$

Short Answer

Expert verified
The expression simplifies to \(\frac{1}{x^{9s}}\).

Step by step solution

01

Understand the Expression

We are given the expression \( \left(x^{-3s}\right)^3 \). This is a power of a power problem involving exponents. Our goal is to simplify it.
02

Apply the Power of a Power Rule

The power of a power rule states that \((a^m)^n = a^{m \cdot n}\). Applying this rule to \( \left(x^{-3s}\right)^3 \), we multiply the exponents: \((-3s) \times 3 = -9s\). Now the expression simplifies to \(x^{-9s}\).
03

Simplify the Negative Exponent

The expression \(x^{-9s}\) can be simplified using the rule that \(a^{-m} = \frac{1}{a^m}\). Therefore, \(x^{-9s} = \frac{1}{x^{9s}}\).

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.

Power of a Power Rule
In mathematics, the power of a power rule is a fundamental principle when working with exponents. The rule states that when you raise a power to another power, you multiply the exponents together. In a mathematical form, this is written as \[ (a^m)^n = a^{m \cdot n} \]where \( a \) is the base, \( m \) is the initial exponent, and \( n \) is the power to which the initial exponent is raised.
  • This rule is particularly useful when you have nested exponents, like \( (x^{m})^{n} \).
  • It simplifies the expression to a single exponent, which can make calculations easier to manage.
  • When applying the power of a power rule, ensure that you correctly multiply the exponents.
For example, in the expression \( (x^{-3s})^3 \), the power of a power rule is applied by multiplying \(-3s\) by \(3\), resulting in an exponent of \(-9s\). This crucial step transforms the complex nested exponent into a more straightforward form, simplifying further calculations.
Negative Exponents
Negative exponents might seem a little daunting at first, but they have a straightforward concept behind them. Simply put, a negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent.
  • If you have \( a^{-m} \), it is equal to \( \frac{1}{a^m} \).
  • This concept reverses the direction of the exponent, essentially flipping the fraction.
  • It's an important rule because it helps convert seemingly complex expressions into fractional forms that are often easier to understand or use in calculations.
So, for our simplified expression \( x^{-9s} \), we utilize the negative exponent rule to convert it to \( \frac{1}{x^{9s}} \). This transformation means that the base \( x \), when raised to \( 9s \), is inverted, making the expression a fraction.
Simplifying Expressions
Simplifying expressions is an essential skill in algebra that helps in reducing mathematical expressions to their simplest form. The goal is to find a way to represent the expression in the most efficient, clear, and concise manner, ensuring that all parts are reduced while maintaining equivalency.
  • It involves using a variety of algebraic rules and properties, such as combining like terms, using exponent rules, and simplifying fractions.
  • Simplifying makes it easier to interpret expressions, perform calculations, and solve equations.
  • In our example, the expression \( (x^{-3s})^3 \) is first simplified by applying the power of a power rule, leading to \( x^{-9s} \).
  • Then, recognizing the negative exponent, it is further simplified to \( \frac{1}{x^{9s}} \).
By breaking down each step, we ensure a comprehensive understanding of how to strategically apply algebraic rules, resulting in a streamlined and efficient expression.

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

Divide If the divisor contains 2 or more terms, use long division. $$ \frac{y^{3}+3 y^{2}+4}{y-2} $$

Write each number in standard notation. See Example 6 The distance light travels in 1 year is \(9.460 \times 10^{12}\) kilometers. Write this number in standard notation.

Multiply. See Section 5.1 $$ -7 x(x) $$

Multiply. See Section 5.1 $$ 10 x^{2}\left(20 x y^{2}\right) $$

Divide. $$ \frac{x^{6}-x^{4}}{x^{3}+1} $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Calculus

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.