/*! 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 74 Write \(\left(4 x^{3} y^{-4}\rig... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 74

Write \(\left(4 x^{3} y^{-4}\right)^{-2}\) so that only positive exponents appear.

Short Answer

Expert verified
Question: Rewrite the expression \(\left(4 x^{3} y^{-4}\right)^{-2}\) with only positive exponents. Answer: \(\frac{y^8}{16x^6}\)

Step by step solution

01

Apply Power Rule to Exponents

Using the power rule \((a^m)^n = a^{mn}\), apply the exponent \(-2\) to both \(4\), \(x^3\), and \(y^{-4}\) inside the parentheses: \((4 x^{3} y^{-4})^{-2} = 4^{-2}(x^{3})^{-2}(y^{-4})^{-2}\).
02

Simplify Exponents

Now, multiply the exponents, and simplify the expression: \(4^{-2}(x^{3})^{-2}(y^{-4})^{-2} = 4^{-2}x^{-6}y^8\).
03

Convert Negative Exponents to Fractions

For any negative exponent, we apply the rule: \(a^{-m} = \frac{1}{a^m}\). Convert the negative exponents to their reciprocal forms: \(4^{-2}x^{-6}y^8 = \frac{1}{4^2}\cdot\frac{1}{x^6}\cdot y^8\).
04

Simplify the Fraction

Now, simplify the fraction and multiply the terms together: \(\frac{1}{4^2}\cdot\frac{1}{x^6}\cdot y^8 = \frac{y^8}{16x^6}\). So, the expression \(\left(4 x^{3} y^{-4}\right)^{-2}\) with only positive exponents is: \(\frac{y^8}{16x^6}\).

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 Rule
In the world of exponents, the power rule is a handy tool used to simplify expressions. The power rule states that when you have an exponent raised to another exponent, you can multiply the powers together. This can be expressed as \((a^m)^n = a^{mn}\).
For instance, if you start with \((x^3)^2\), using the power rule, you multiply the exponents: \(3 \times 2\), giving \(x^6\).
This rule helps in reducing complex exponentiation into a simpler form and is particularly useful in algebraic expressions involving powers.
Negative Exponents
Negative exponents can seem confusing at first, but understanding them is much simpler than it appears. When dealing with negative exponents, remember that they represent a reciprocal. The rule is: \(a^{-m} = \frac{1}{a^m}\). This means that any base with a negative exponent can be rewritten as one over that base with a positive exponent.

For example, \(x^{-3}\) becomes \(\frac{1}{x^3}\). This concept allows us to transform expressions so that all exponents become positive, making them easier to understand and solve.
Simplifying Expressions
Simplifying expressions is all about making them as straightforward as possible. This often involves using rules like the power rule and handling negative exponents. The goal is to rewrite the expression so that it's easier to work with while maintaining its value.

In our exercise, simplifying involved converting any negative exponents to their positive counterparts and combining terms where possible.
  • Multiply the powers when using the power rule.
  • Change negative exponents to fractions to simplify.
With these steps, complex expressions become much more manageable.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations. They are a foundational part of algebra, and understanding them is crucial for solving equations and simplifying expressions.

In our example, the expression \((4x^3y^{-4})^{-2}\) is an algebraic expression that involves variables \(x\) and \(y\) raised to different powers. Simplifying it using rules of exponents transforms it into a more straightforward form.
Often, these expressions can be manipulated to find solutions or to make calculations easier, reflecting the beauty and utility of algebra.

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

For the following problems, perform the divisions. $$ \frac{x^{2}-x-6}{x^{2}-2 x-3} $$

For the following problems, solve the rational equations. Solve \(\frac{1}{R}=\frac{1}{E}+\frac{1}{F}\) for \(E\).

For the following problems, divide the polynomials. $$ a^{2}+9 a+18 \text { by } a+3 $$

For the following problems, divide the polynomials. $$ c^{2}+3 c-88 \text { by } c-8 $$

For the following problems, perform the indicated operations. $$ \frac{8 a-12}{3 a+3} \div \frac{(a+1)^{2}}{4 a-6} $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.