/*! 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 86 In Exercises 85-94, factor and s... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 86

In Exercises 85-94, factor and simplify each algebraic expression. $$x^{3 / 4}-x^{1 / 4}$$

Short Answer

Expert verified
The factored and simplified form of \(x^{3 / 4}-x^{1 / 4}\) is \(x^{1/4}(x^{1/2} - 1)\)

Step by step solution

01

Identify the Common Factor

In \(x^{3 / 4}-x^{1 / 4}\), it can be noticed that both terms have \(x^{1 / 4}\) in common. Therefore, this can be factored out.
02

Factor out the Common Term

Factoring out the common term \(x^{1 / 4}\) from \(x^{3 / 4}-x^{1 / 4}\) gives \(x^{1 / 4}(x^{1 / 2} - 1)\). Here, the remaining term \(x^{1 / 2}\) is obtained by subtracting the exponent 1/4 from 3/4, according to the rule \(x^{a}-x^{b}=x^{a-b}\).
03

Simplify the Expression

The expression is already in its simplest form since there are no more common factors to be factored out.

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.

Common Factor in Algebra
In algebra, identifying a common factor is a key step in simplifying expressions. A common factor is essentially a term that is present in every part of the algebraic expression. In the expression \(x^{3/4} - x^{1/4}\), both terms have \(x^{1/4}\) as a common factor. This means that each term can be divided by \(x^{1/4}\).
  • The common factor is the smallest power of a variable or term that divides each term evenly.
  • Identifying a common factor helps in breaking down the expression into simpler components.
Once the common factor is identified, it is factored out of the expression. This process of factoring helps to reduce the complexity of solving the expression by dealing with smaller and simpler components instead of the whole at once.
Simplifying Expressions
Simplifying expressions is an important skill in algebra, aimed at making an expression easier to work with by reducing it to its simplest form. When you simplify an expression like \(x^{1/4}(x^{1/2} - 1)\), you're ensuring that there are no further common factors that can be extracted or reduced.
  • Check for common factors in each term within the brackets after factoring out any common term.
  • Combine like terms where possible.
  • Keep expressions as tidy as possible, removing any unnecessary components.
To assess whether an expression is fully simplified, ensure that there are no further options for factoring or reducing coefficients and variable powers.
Exponent Rules in Algebra
Understanding exponent rules is crucial for correctly simplifying expressions and managing algebraic terms with powers. In the expression \(x^{3/4} - x^{1/4}\), exponent rules help us understand how to manipulate and reduce powers when a common factor is present.
  • According to exponent rules, when dividing terms with the same base, subtract the exponents: \(x^{a} / x^{b} = x^{a-b}\).
  • For multiplication, add exponents: \(x^{a} \cdot x^{b} = x^{a+b}\).
  • Apply these rules systematically to simplify complicated expressions.
By factoring \(x^{1/4}\) out of \(x^{3/4} - x^{1/4}\), you effectively use the rule \(x^{a} - x^{b} = x^{a-b}\) to find that the remaining term is \(x^{1/2}\). This reduction helps in clearly showing the expression鈥檚 structure and simplifies further calculations.

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

state the name of the property illustrated. $$ 6 \cdot(2 \cdot 3)=6 \cdot(3 \cdot 2) $$

state the name of the property illustrated. $$ \frac{1}{(x+3)}(x+3)=1, x \neq-3 $$

Use the quotient rule to simplify the expressions in Exercises \(17-26 .\) Assume that \(x>0\) $$\sqrt{\frac{49}{16}}$$

Simplify each exponential expression $$ \frac{20 x^{24}}{10 x^{6}} $$

simplify each algebraic expression. $$ -(-14 x) $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

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.