/*! 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 45 Simplify the expression. $$ ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 45

Simplify the expression. $$ \frac{3 a+1}{a^{7}} \div \frac{a+1}{3 a^{8}} $$

Short Answer

Expert verified
The simplified expression is \( \frac{3a^2 + 3a}{a^7 + a^8} \).

Step by step solution

01

Rewrite the Division as Multiplication

To simplify the expression \( \frac{3a + 1}{a^7} \div \frac{a + 1}{3a^8} \), we first rewrite the division as multiplication by the reciprocal. This changes the expression to \( \frac{3a + 1}{a^7} \times \frac{3a^8}{a + 1} \).
02

Simplify by Cancelling Common Factors

Next, we examine the factors in the numerator and denominator. Notice that \((a+1)\) and \(3a + 1\) are unique and cannot be canceled directly, but the \(a^7\) in the denominator of the first fraction can be partially canceled with \(a^8\) in the numerator of the reciprocal, leaving \(a\) in the numerator. Thus, the expression becomes \( \frac{(3a + 1) \cdot 3a}{(a^7) \cdot (a + 1)} \).
03

Rearrange and Finalize the Simplification

Combine the numerators and denominators, factoring out and simplifying wherever possible. The expression becomes \( \frac{3a (3a + 1)}{a^7 (a + 1)} \). Since no further terms can be canceled, the expression is now fully simplified to \( \frac{3a^2 + 3a}{a^7 + a^8} \), recognizing that \(a\) and \((a + 1)\) remain in separate factors.

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.

Simplification
Simplification of algebraic expressions, especially involving fractions, is about reducing them to their simplest forms. This involves:
  • Identifying common terms in both the numerator and the denominator that can be simplified or canceled.
  • Looking for greatest common factors and terms that repeat.
In the exercise, we have an expression that requires simplification, beginning with rewriting division as multiplication, which can sometimes help clarify which terms may cancel more easily. By examining each term carefully, you ensure nothing important is left out while simplifying the expression. Typically, a simplified expression is much easier to solve or integrate into more complex equations, making this step particularly crucial in algebra.
Division of Fractions
When dividing fractions, the key is to understand that division can be transformed into multiplication. This is achieved by
  • Turning the division sign into a multiplication sign.
  • Flipping the second fraction (also known as taking its reciprocal).
For instance, dividing by \( \frac{a+1}{3a^8} \) involves flipping it to \( \frac{3a^8}{a+1} \) and then multiplying. This method isn't just limited to numbers - it works perfectly well with variables and algebraic terms too! By converting the division into a multiplication problem, you often make it easier to cancel and simplify terms, which paves the path for the simpler expression that follows.
Multiplication of Fractions
Multiplying fractions is straightforward once you've got the hang of the basic steps:
  • Multiply the numerators together to get the new numerator.
  • Multiply the denominators together to get the new denominator.
In this operation, it's essential to keep track of all terms. For algebraic fractions, this means carefully working with each variable and coefficient. The trickier aspect often involves managing the variables, especially when they come with exponents, like \(a^7\) and \(a^8\). After multiplying, look for spots where you can factor and simplify further鈥攐ften this means canceling out common terms in the numerator and denominator. This step reduces complex fractions to their simplest terms, which can significantly simplify future computations.

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

Apply the distributive property. $$(y-3) 6 y$$

Multiply the expressions. $$(2-3 x)^{2}$$

Multiply the binomials. $$(-2 x+3)(x-2)$$

Add the polynomials. $$\left(z^{4}-6 z^{2}+3\right)+\left(5 z^{3}+3 z^{2}-3\right)$$

Multiply the binomials. $$\left(x^{2}-2\right)\left(x^{2}+4\right)$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

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.