/*! 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 22 Simplify each of the following a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 22

Simplify each of the following as completely as possible. $$\frac{z^{2} z^{7}}{z^{4} z^{6}}$$

Short Answer

Expert verified
\( \frac{1}{z} \)

Step by step solution

01

- Combine the exponents in the numerator

Combine the exponents of the terms in the numerator by adding them: \(z^2 z^7 = z^{2+7} = z^9\)
02

- Combine the exponents in the denominator

Combine the exponents of the terms in the denominator by adding them: \(z^4 z^6 = z^{4+6} = z^{10}\)
03

- Divide the exponents

Use the property of exponents \( \frac{a^m}{a^n} = a^{m-n} \) to simplify the fraction: \( \frac{z^9}{z^{10}} = z^{9-10} = z^{-1} \)
04

- Simplify further if needed

A negative exponent indicates the reciprocal, so: \(z^{-1} = \frac{1}{z}\)

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.

Combining Exponents
When simplifying expressions with exponents, it's common to see terms that need to be combined. Let's break down the process. If you have the same base, you can combine the exponents by adding them. For instance, in the numerator, we have \(z^2\) and \(z^7\). Since they have the same base \(z\), we add their exponents to get \(z^{2+7} = z^9\). Similarly, in the denominator, we combine \(z^4\) and \(z^6\) to get \(z^{4+6} = z^{10}\). This step is crucial as it simplifies the expression significantly and sets the stage for further simplification.
Properties of Exponents
Understanding the fundamental properties of exponents is essential for simplifying and manipulating expressions. One important property is that when you divide like bases, you subtract the exponents from one another: \(\frac{a^m}{a^n} = a^{m-n}\). Applying this to our example, we get \(\frac{z^9}{z^{10}} = z^{9-10} = z^{-1}\). This property is incredibly useful for simplifying complex fractions involving exponents.
Negative Exponents
Negative exponents can be a bit confusing at first, but they are quite manageable once you understand their meaning. A negative exponent indicates that the base is on the 'wrong' side of the fraction line. In other words, \(z^{-1}\) is the same as \(\frac{1}{z}\). This tells us that we need to take the reciprocal of the base. Therefore, our simplified form of \(z^{-1}\) becomes \(\frac{1}{z}\). This concept helps in converting expressions to a more familiar and often more usable form.
Reciprocal
The notion of a reciprocal is closely linked to negative exponents. The reciprocal of a number is simply \(1\) divided by that number. For example, the reciprocal of \(z\) is \(\frac{1}{z}\). When facing a term with a negative exponent, such as \(z^{-1}\), we use the concept of the reciprocal to rewrite it as \(\frac{1}{z}\). This conversion is particularly helpful when you're looking to simplify expressions or solve equations involving exponents. By understanding and applying the concept of reciprocals, you can turn potentially complex expressions into more straightforward ones.

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

A car rental agency charges \(\$ 15\) per day plus \(\$ 0.20\) per mile. If a 3 -day rental cost \(\$ 72,\) how many miles were driven?

Convert each number into standard notation. $$4.29 \times 10^{-7}$$

Solve each of the following problems algebraically. Be sure to label what the variable represents. A large, full tank contains several liquids. If one fifth of the tank is water, 5 gallons is orange juice, and three quarters of the tank is wine, how many gallons of wine are in the tank?

Estimate the answer without actually carrying out the computation and make the most appropriate choice. If you divide \(7.92 \times 10^{-5}\) by \(2.21 \times 10^{-2},\) the result is closest to (a) 100 (b) 1000 (c) 0.1 (d) 0.01 (e) 0.001

The length of a rectangle is 3 more twan twice its width. Using \(W\) as the variable, express the perimeter and area of the rectangle in terms of \(W\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

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.