Problem 27 $\frac{32 b^{4}}{-12 b^{3}}$... [FREE SOLUTION]

91影视

Elementary Algebra Within Reach

Ron Larson, Kimberly Nolting

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 5: Problem 27

$\frac{32 b^{4}}{-12 b^{3}}$

Short Answer

Expert verified

Simplified form of the given expression $\frac{32 b^{4}}{-12 b^{3}}$ is $-\frac{2^{4}}{3} * b$.

Step by step solution

Factorize Numerator and Denominator

Factorize 32 and -12, we get 32 = 2^5 and -12 = -2^2*3. Also, the exponent rules says, for any non-zero number, a with integers m, n, $a^{m}/a^{n} = a^{m-n}$. So, applying this for b, we get $b^{4}/b^{3} = b^(4-3)$ which equals b. So we can write the given expression as $(-2) * (\frac{2^{5}}{2^{2} * 3}) * b$.

Simplify expression

Simplify the expression we get from Step 1. We use the same exponent rules to simplify $\frac{2^{5}}{2^{2}} = 2^{5-2} = 2^{3}$. Then replace $\frac{2^{5}}{2^{2} * 3}$ with $2^{3}/3$ in the expression we get from Step 1, we get -2* $2^{3}/3$ * b.

Write the final answer

The final simplified expression is $-\frac{2^{4}}{3} * b$.

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

When working with algebraic expressions, understanding exponent rules is crucial. Exponents indicate how many times a number is multiplied by itself. In our exercise, we used the rule: $a^{m}/a^{n} = a^{m-n}$. This means that when you divide like bases, you can subtract the exponents. Applying this in the given problem, we changed $b^{4}/b^{3}$ into $b^{(4-3)}$, which simplifies to $b$. Using exponent rules can help simplify expressions drastically, transforming complex problems into easier ones.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form without changing their value. In the given exercise, after factoring, we started the simplification by applying the exponent rules to the numeral expressions as well. We found $\frac{2^{5}}{2^{2}}$, which simplifies further to $2^{3}$, because you subtract the exponents: $5 - 2 = 3$. From $-2 \cdot \frac{2^{5}}{2^{2} \cdot 3} \cdot b$, we can replace it with $-2 \cdot \frac{2^{3}}{3} \cdot b$. The key is performing operations step by step, one piece at a time, to avoid overloading the process.

Factoring

Factoring is the process of breaking down an expression into the product of simpler terms or factors. It helps to make the expression easier to work with and can often reveal simplifications not previously obvious. In tackling our problem, the first step was to factorize the numbers in the numerator and denominator. For 32, we noted it as $2^5$ and for $-12$ as $-2^2 \cdot 3$. By rewriting numbers and terms in this factored form, we managed not only to simplify the fraction but also to clearly see the operation of the exponent rules. Always start factoring by identifying common factors, which will make simplifying later steps more efficient.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

\(\frac{32 b^{4}}{-12 b^{3}}\)

Short Answer

Step by step solution

Factorize Numerator and Denominator

Simplify expression

Write the final answer

Key Concepts

Exponent Rules

Simplifying Expressions

Factoring

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Decision Maths

Mechanics Maths

Discrete Mathematics

Logic and Functions

Probability and Statistics

Study anywhere. Anytime. Across all devices.