/*! 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 116 Simplify. Do not use negative ex... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 116

Simplify. Do not use negative exponents in the answer. \(\frac{d\left(d^{-3}\right)^{-3}}{d^{-7}}\)

Short Answer

Expert verified
The simplified expression is \(d^{17}\).

Step by step solution

01

Simplify the Power of a Power

The expression in the numerator is \((d^{-3})^{-3}\). Apply the power of a power property of exponents: \((a^m)^n = a^{m \times n}\). Therefore, \((d^{-3})^{-3} = d^{-3 \times -3} = d^9\).
02

Simplify the Fraction

The expression now is \(\frac{d \cdot d^9}{d^{-7}}\). Apply the product of powers property: \(d^a \cdot d^b = d^{a+b}\). This gives \(d \cdot d^9 = d^{1+9} = d^{10}\).
03

Combine the Exponents

Rewrite the expression as \(\frac{d^{10}}{d^{-7}}\). Use the quotient of powers property: \(\frac{a^m}{a^n} = a^{m-n}\). Therefore, \(d^{10-(-7)} = d^{10+7} = d^{17}\).
04

Final Result

The expression \(\frac{d\left(d^{-3}\right)^{-3}}{d^{-7}}\) simplifies to \(d^{17}\). This is the final result without negative exponents.

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.

Exponents
Exponents are a way to express repeated multiplication of a number or expression. The base is the number being multiplied, while the exponent, often displayed as a small number to the right of the base, indicates how many times the base is multiplied by itself. For example, in the expression \(d^3\), \(d\) is the base and \(3\) is the exponent, meaning \(d\) is multiplied by itself three times: \(d \times d \times d\).
  • Exponents simplify the way we express large numbers or complex operations.
  • Any number raised to the power of one is itself.
  • Any non-zero number raised to the power zero equals one.
Understanding how to manipulate exponents is crucial for simplifying algebraic expressions, especially when other properties such as the Power of a Power Property or Product of Powers Property come into play.
Power of a Power Property
The Power of a Power Property is a fundamental principle of exponents that helps simplify expressions where an exponent is raised to another exponent. It states: \((a^m)^n = a^{m \times n}\). Essentially, you multiply the exponents together while keeping the base constant. Consider the expression \((d^{-3})^{-3}\). Applying this property, we multiply the exponents:
  • Calculate \(-3 \times -3\).
  • This results in \(9\), so \((d^{-3})^{-3} = d^{9}\).
Using this property simplifies complex exponentiation, reducing multi-layered exponents to a single, manageable one, and makes calculations much easier.
Product of Powers Property
The Product of Powers Property allows us to simplify expressions where the same base is being multiplied with different exponents. This property is articulated as: \(a^m \cdot a^n = a^{m+n}\). Essentially, it tells us to add the exponents when multiplying like bases. For example, given \(d \cdot d^9\):
  • We recognize both terms share the base \(d\).
  • Add the exponents: \(1 + 9 = 10\).
  • So, \(d \cdot d^9 = d^{10}\).
This property streamlines calculations and expression manipulation by consolidating multiple terms into a simpler form.
Quotient of Powers Property
The Quotient of Powers Property is a useful tool when simplifying expressions that involve division of like bases with exponents. This property states: \(\frac{a^m}{a^n} = a^{m-n}\). Essentially, you subtract the exponent in the denominator from the exponent in the numerator. Let's apply this to the expression \(\frac{d^{10}}{d^{-7}}\):
  • Both numerator and denominator have the base \(d\).
  • Subtract the exponents: \(10 - (-7) = 17\).
  • This simplifies to \(d^{17}\).
Utilizing this property makes dealing with fraction-like expressions simpler by reducing them to single exponents, facilitating easier computation and streamlined results.

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

Perform each division. $$ \left(12 d^{2}-20 d+3\right) \div(6 d-1) $$

Perform each division. $$ \frac{-18 w^{6}-9}{9 w^{4}} $$

WRITING Explain why we can simplify \(x^{4} \cdot x^{5},\) but cannot simplify \(x^{4}+x^{5}\)

Perform the operations. $$ \left(6 b+\frac{1}{2}\right)\left(6 b-\frac{1}{2}\right) $$

Perform each division. $$ \frac{x^{3}-8}{x-2} $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

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.