/*! 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 42 Simplify by writing each express... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 42

Simplify by writing each expression with positive exponents. Assume that all variables represent nonzero real numbers. $$ \frac{y^{4}}{y^{-6}} $$

Short Answer

Expert verified
The simplified expression is \(y^{10}\).

Step by step solution

01

Understand the Problem

The given expression is \(\frac{y^{4}}{y^{-6}}\). The goal is to simplify this expression by writing it with positive exponents only.
02

Apply the Law of Exponents for Division

Use the rule \(\frac{a^m}{a^n} = a^{m-n}\). Here, divide the exponents by subtracting the exponent in the denominator from the exponent in the numerator: \(\frac{y^{4}}{y^{-6}} = y^{4 - (-6)} = y^{4 + 6}\).
03

Simplify the Exponential Expression

Combine the exponents: \(y^{4 + 6} = y^{10}\). This gives us the simplified expression with a positive exponent.

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.

positive exponents
Positive exponents are used to indicate how many times a number, called the base, is multiplied by itself. For example, in the expression \(y^4\), the base \(y\) is multiplied by itself four times: ewline ewline \( y^4 = y \times y \times y \times y\) . ewline ewline Using only positive exponents in our final expressions makes them clearer and more standardized. This is why converting negative exponents into positive ones is an essential skill in algebra.
laws of exponents
The laws of exponents are rules that describe how to handle and simplify expressions involving exponents. These rules are crucial when working with exponential expressions. Here are a few key laws:ewline ewline - Product of Powers: \(a^m \times a^n = a^{m+n}\) ewline - Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\) ewline - Power of a Power: \((a^m)^n = a^{m \times n}\) ewline ewline In our exercise, we mainly used the 'Quotient of Powers' rule to simplify the expression \(\frac{y^4}{y^{-6}}\).
exponential expressions
An exponential expression is a mathematical phrase that includes a base raised to an exponent. These expressions are used frequently in algebra and other areas of mathematics. An exponential expression looks something like \(b^n\), where \(b\) is the base and \(n\) is the exponent.ewline When simplifying exponential expressions, it is important to follow the laws of exponents. In the solution, we started with \(\frac{y^4}{y^{-6}}\) and applied the Quotient of Powers rule to get \(y^{4 - (-6)}\) which simplifies to \(y^{10}\).
algebra
Algebra involves the study of symbols and the rules for manipulating these symbols. It is a foundational branch of mathematics that deals with the relationships between numbers and variables. Basic algebra often includes operations such as addition, subtraction, multiplication, and division with both real numbers and variables.ewline In our exercise, we used algebraic techniques to simplify an exponential expression involving the variable \(y\).ewline ewline By understanding and applying algebraic rules like the laws of exponents, students can simplify complex expressions and solve equations effectively.

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

Each statement comes from Astronomy! A Brief Edition by James B. Kaler (Addison-Wesley). If the number in boldface italics is in scientific notation, write it without exponents. If the number is written without exponents, write it in scientific notation. (IMAGE CANNOT COPY). At maximum, a cosmic ray particle - a mere atomic nucleus of only \(10^{-13} \mathrm{cm}\) across- can carry the energy of a professionally pitched baseball. (page \(445)\).

Our system of numeration is called a decimal system. In a whole number such as 2846 each digit is understood to represent the number of powers of 10 for its place value. The 2 represents two thousands \(\left(2 \times 10^{3}\right),\) the 8 represents eight hundreds \(\left(8 \times 10^{2}\right),\) the 4 represents four tens \(\left(4 \times 10^{1}\right),\) and the 6 represents six ones (or units) \(\left(6 \times 10^{\circ}\right)\) \(2846=\left(2 \times 10^{3}\right)+\left(8 \times 10^{2}\right)+\left(4 \times 10^{1}\right)+\left(6 \times 10^{0}\right) \quad\) Expanded form $$ \text {Keeping this information in mind,} $$ Divide the polynomial \(2 x^{3}+8 x^{2}+4 x+6\) by 2

The special product $$ (x+y)(x-y)=x^{2}-y^{2} $$ can be used to perform some multiplication problems. Here are two examples. $$ \begin{aligned} 51 \times 49 &=(50+1)(50-1) \\ &=50^{2}-1^{2} \\ &=2500-1 \\ &=2499 \end{aligned} \quad | \begin{aligned} 102 \times 98 &=(100+2)(100-2) \\ &=100^{2}-2^{2} \\ &=10,000-4 \\ &=9996 \end{aligned} $$ Once these patterns are recognized, multiplications of this type can be done mentally. Use this method to calculate each product mentally. $$ 103 \times 97 $$

In theory there are \(1 \times 10^{9}\) possible Social Security numbers. The population of the United States is about \(3 \times 10^{8} .\) How many Social Security numbers are available for each person? (Source: U.S. Census Bureau.)

Write each product as a sum of terms. Write answers with positive exponents only. Simplify each term. $$ \frac{1}{2 p}\left(4 p^{2}+2 p+8\right) $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Geometry

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.