/*! 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 50 Evaluate each expression. $$r^... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 50

Evaluate each expression. $$r^{2} \div s^{2} \text { if } r=-\frac{3}{4} \text { and } s=1 \frac{1}{3}$$

Short Answer

Expert verified
The expression evaluates to \( \frac{81}{256} \).

Step by step solution

01

Substitute the Values

First, substitute the given values of \( r \) and \( s \) into the expression. We have \( r = -\frac{3}{4} \) and \( s = 1 \frac{1}{3} \). Substitute these into the expression: \( \left(-\frac{3}{4}\right)^{2} \div \left(1 \frac{1}{3}\right)^{2} \).
02

Simplify \( s \) into an Improper Fraction

Convert the mixed number \( s = 1 \frac{1}{3} \) into an improper fraction. \( 1 \frac{1}{3} = \frac{3 \times 1 + 1}{3} = \frac{4}{3} \). Now the expression becomes \( \left(-\frac{3}{4}\right)^{2} \div \left(\frac{4}{3}\right)^{2} \).
03

Calculate \( r^{2} \)

Find the square of \( r \). We have \( r = -\frac{3}{4} \), so \( \left(-\frac{3}{4}\right)^{2} = \frac{(-3)^{2}}{4^{2}} = \frac{9}{16} \).
04

Calculate \( s^{2} \)

Find the square of \( s \). We have \( s = \frac{4}{3} \), so \( \left(\frac{4}{3}\right)^{2} = \frac{4^{2}}{3^{2}} = \frac{16}{9} \).
05

Calculate the Division

Now perform the division: \( \frac{9}{16} \div \frac{16}{9} \). Dividing by a fraction involves multiplying by its reciprocal. Thus, \( \frac{9}{16} \div \frac{16}{9} = \frac{9}{16} \times \frac{9}{16} = \frac{81}{256} \).

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 Manipulation
Fraction manipulation is a skill that helps in working with parts of a whole or representing numbers out of a particular total. This involves several sub-skills:
  • Simplifying: Fractions can often be simplified by dividing the numerator and the denominator by their greatest common divisor.
  • Converting: Mixed numbers need to be converted to improper fractions to simplify calculations. For instance, converting a number like \(1 \frac{1}{3}\) into \(\frac{4}{3}\) ensures uniformity in mathematical operations.
  • Rewriting: Operations often require rewriting fractions. For example, converting multiplication or division into straightforward expressions makes them easier to handle.
This exercise involves changing mixed numbers to improper fractions and correctly preparing them for further calculations, such as exponentiation or division.
Exponentiation
Exponentiation is a mathematical process where a number is multiplied by itself a certain number of times. This exercise asks us to understand exponentiation involving fractions:
  • Squaring: Squaring a number involves multiplying it by itself. For instance, \(r^2\), where \(r = -\frac{3}{4}\), means \((-\frac{3}{4}) \times (-\frac{3}{4}) = \frac{9}{16}\).
  • Fraction Exponentiation: When exponents apply to fractions, apply the power to both the numerator and the denominator separately. For example, \(\left(\frac{4}{3}\right)^2 = \frac{4^2}{3^2} = \frac{16}{9}\).
It's key to treat the operation on both parts of the fraction equally. This ensures correctness regardless of whether dealing with integers or fractions in calculations.
Division of Fractions
Dividing fractions may sound complex, but a simple rule makes it straightforward: invert and multiply.
  • Reciprocal Understanding: To divide by a fraction, multiply by its reciprocal. The reciprocal of \(\frac{16}{9}\) is \(\frac{9}{16}\).
  • Applying the Rule: If our goal is to compute \(\frac{9}{16} \div \frac{16}{9}\), convert this to \(\frac{9}{16} \times \frac{9}{16}\).
  • Multiplication: Multiply the numerators together and the denominators together, getting \(\frac{81}{256}\).
This technique of inverting and multiplying means that dividing by complex fractions becomes a simple multiplication task, making it easier to handle any problems involving division of fractions.

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

Estimate each product. $$-5 \frac{1}{3} \cdot 3 \frac{4}{5}$$

Find the GCF of each pair of monomials. $$4 p^{2}, 6 p$$

Write each improper fraction as a mixed number in simplest form. $$\frac{9}{4}$$

Solve each equation. Check your solution. $$\frac{5}{8}=-\frac{1}{2} r$$

Write each improper fraction as a mixed number in simplest form. $$\frac{24}{5}$$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

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.