/*! 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 103 Raising a Product or a Quotient ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 103

Raising a Product or a Quotient to a Power $$ \left(-2 a^{-2} b\right)^{-3} $$

Short Answer

Expert verified
-\frac{a^{6}}{8 b^{3}}

Step by step solution

01

Expand the Expression

Expand the product inside the parentheses. The expression can be expanded to \[-2^{-3} (a^{-2})^{-3} b^{-3} \].
02

Simplify Each Component

Simplify each part of the expanded expression: \( -2^{-3} = \frac{1}{(-2)^3} = -\frac{1}{8} \), \( (a^{-2})^{-3} = a^{6} \), and \( b^{-3} = \frac{1}{b^3} \).
03

Combine the Simplified Parts

Combine all the simplified parts: \[-\frac{1}{8} \times a^{6} \times \frac{1}{b^{3}} = -\frac{a^{6}}{8 b^{3}} \].

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.

Exponent Rules
Exponents are a fundamental part of algebra that help us compactly represent repeated multiplication. It's essential to grasp the different rules governing exponents to simplify expressions effectively. Here are some key exponent rules to remember:

  • \textbf{Product of Powers Rule}: To multiply powers with the same base, add their exponents, e.g., \(a^m \times a^n = a^{m+n}\).
  • \textbf{Power of a Power Rule}: To raise a power to another power, multiply the exponents, e.g., \((a^m)^n = a^{m \times n}\).
  • \textbf{Power of a Product Rule}: To raise a product to a power, raise each factor of the product to that power, e.g., \((ab)^n = a^n \times b^n\).
  • \textbf{Power of a Quotient Rule}: To raise a quotient to a power, raise both the numerator and the denominator to that power, e.g., \(\frac{a}{b})^n = \frac{a^n}{b^n}\).
  • \textbf{Negative Exponent Rule}: A negative exponent means reciprocal, e.g., \(a^{-n} = \frac{1}{a^n}\).
In the given exercise, we apply these rules to simplify the expression \(\text{-} 2 a^{-2} b)^{-3}\). By using these rules effectively, we break down complex expressions into more manageable parts.
Simplifying Expressions
Simplifying expressions allows us to rewrite them in a more compact and understandable form. It often involves combining like terms, using exponent rules, and reducing fractions. Simplifying an expression can make it easier to solve equations or understand mathematical relationships.

For the given exercise, we started with the expression \( \text{-} 2 a^{-2} b^{-3} \). Here's the simplified approach:

  • Rewrite \( -2^{-3}\) as \( \frac{1}{-8}\).
  • For \( (a^{-2})^{-3}\), apply the Power of a Power rule: \( a^{-2 \times -3} = a^{6}\).
  • Rewrite \( b^{-3} \) as \( \frac{1}{b^3} \).
  • This simplified approach helps break down the complex expression into simpler parts, making the whole process more manageable.
    • Algebraic Fractions
    Algebraic fractions are fractions that contain algebraic expressions in their numerators or denominators. Simplifying algebraic fractions involves reducing the expression to its most basic form by applying exponent rules, factoring, and cancelling common factors.

    In our exercise, we encounter the algebraic fraction \(-\frac{a^{6}}{8 b^{3}}\). Here are the steps to deal with such fractions:

    By simplifying algebraic fractions, we make the expressions easier to analyze and solve. Understanding and practicing these steps will help you handle more complex algebraic fractions with confidence.

    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

    Find an equivalent expression by factoring. $$ a b+b $$

    Explain in your own words why \(7 / 0\) is undefined.

    The height and sides of a triangle are four consecutive integers. The height is the first integer, and the base is the third integer. The perimeter of the triangle is 42 in. Find the area of the triangle.

    Jasmine is paid \(\$ 500\) per week plus \(10 \%\) of all sales she makes. a) Let \(x\) represent the amount of Jasmine's weekly sales, in dollars, and \(y\) her weekly paycheck. Write an equation expressing \(y\) in terms of \(x\) b) One week, Jasmine earned \(\$ 900 .\) What were her sales that week?

    Calculate using the rules for order of operations. If an expression is undefined, state this. $$ 32-(-5)^{2}+15 \div(-3) \cdot 2 $$
    See all solutions

    Recommended explanations on Math Textbooks

    Decision Maths

    Read Explanation

    Theoretical and Mathematical Physics

    Read Explanation

    Discrete Mathematics

    Read Explanation

    Statistics

    Read Explanation

    Logic and Functions

    Read Explanation

    Pure Maths

    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.