/*! 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 94 \(\left(\frac{5}{4}\right)^{-3}\... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 94

\(\left(\frac{5}{4}\right)^{-3}\)

Short Answer

Expert verified
\(\left(\frac{5}{4}\right)^{-3} = \frac{64}{125}\)

Step by step solution

01

Apply the rule of negative exponents

Replace \( \left(\frac{5}{4}\right)^{-3}\) with \(\frac{1}{\left(\frac{5}{4}\right)^3}\) according to the rule of negative exponents.
02

Simplify the denominator

Calculate \(\left(\frac{5}{4}\right)^3 = \left(\frac{5}{4}\right) * \left(\frac{5}{4}\right) * \left(\frac{5}{4}\right) = \frac{125}{64}\)
03

Final Simplification

Rewrite the expression \(\frac{1}{\frac{125}{64}}\) as \(\frac{64}{125}\)

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.

Fraction Simplification
Fractions are a way to represent parts of a whole. In our exercise, we deal with simplifying fractions, an essential aspect when working with expressions involving exponents. Let's break it down.

First, understanding how to simplify involves recognizing and transforming complex fractions. Suppose you have a fraction within a fraction, like \( \frac{1}{\frac{a}{b}} \). To simplify, multiply both the numerator and the denominator by \( b \) (the denominator of the smaller fraction), effectively cancelling out \( b \). This results in \( \frac{b}{a} \).

  • The objective is to find an equivalent fraction that is simple and easy to interpret.
  • Always look for common factors you can cancel out.
  • Converting complex fractions to simpler forms helps both readability and further calculations.
Algebraic Expressions
Algebraic expressions involving fractions and exponents can be daunting at first. However, with a clear understanding, these become straightforward. Let's talk about how to interpret and manipulate these expressions.

Algebraic expressions are mathematical phrases that can include numbers, variables (like \( x \)), and operation symbols. In our example, consider the expression \( \left(\frac{5}{4}\right)^{-3} \). This involves both a fractional base and a negative exponent, which we'll manage using certain mathematical rules.

When faced with such expressions, identify:
  • The base of the expression 鈥� here it's the fraction \( \frac{5}{4} \).
  • Operations applied to the base 鈥� like raising the fraction to a power.
  • Potential multiplication or division steps needed for simplification.
Exponent Rules
Exponents are shorthand for repeated multiplication. They can be positive, negative, or zero, each holding a distinct meaning. In algebra, understanding the rules that govern exponents is crucial.

Let's delve into negative exponents, using the rule applied in our solution: when a base is raised to a negative power, you take the reciprocal of the base and change the exponent to positive. This is why \( \left(\frac{5}{4}\right)^{-3} \) becomes \( \frac{1}{\left(\frac{5}{4}\right)^3} \).

The fundamental exponent rules to remember include:

  • Negative Exponent Rule: \( a^{-n} = \frac{1}{a^n} \).
  • Product of Powers Rule: Multiply the same bases by adding exponents; \( a^m \times a^n = a^{m+n} \).
  • Power of a Power Rule: Multiply exponents when raising a power to another power; \( (a^m)^n = a^{m \times n} \).
With these rules, you can confidently simplify and manipulate expressions involving exponents.

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

\((u+5)^{2}+(u-5)^{2}\)

\(\frac{b^{2}-2 b}{b}\)

\(\frac{x^{2}-3 x+2}{x-1}+(4 x-3)\)

\(\frac{-175}{-42}\)

What number is \(25 \%\) of 45 ?
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

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.