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Chapter 0: Problem 94

Factor and simplify each algebraic expression. $$x^{\frac{3}{4}}-x^{\frac{1}{4}}$$

Short Answer

Expert verified
The factored and simplified form of the given expression is \(x^{\frac{1}{4}}(x^{\frac{1}{2}} - 1)\)

Step by step solution

01

Identify Common Factor

First, spot the common factor in the two terms, which is \(x^{\frac{1}{4}}\). Both terms can be divided evenly by this factor.
02

Factor out the common factor

Factor out \(x^{\frac{1}{4}}\) from each term in the expression. Doing this, we get: \(x^{\frac{1}{4}}(x^{\frac{1}{2}} - 1)\). Here, we used the property of exponents that states \(a^n/a^m = a^{n-m}\), so here \(x^{\frac{3}{4}}/x^{\frac{1}{4}} = x^{\frac{3}{4} - \frac{1}{4}} = x^{\frac{1}{2}}\).
03

Final Simplified Form

The expression is now factored and simplified as much as possible, so we finish at this step.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Factor
In algebra, one of the foundational ideas is identifying a common factor in an expression. A common factor is a term that divides each term of the expression evenly. When factoring, it's similar to finding the greatest common denominator but applied to algebraic terms rather than numbers.
For example, consider the expression: \(x^{\frac{3}{4}} - x^{\frac{1}{4}}\). Here, both terms have a common factor of \(x^{\frac{1}{4}}\).
This means that \(x^{\frac{1}{4}}\) can be factored out, as it is present in each part of the expression.
  • Factoring helps in simplifying expressions and solving equations.
  • It can make complex problems easier to manage.
  • Recognizing common factors is a valuable skill in many areas of mathematics.
Exponent Properties
Understanding the properties of exponents is crucial, especially when working with algebraic expressions involving powers. Exponent rules can simplify calculations and help in factoring expressions effectively.
One such property is the rule that \(a^n / a^m = a^{n-m}\), which allows us to manipulate terms efficiently. For our example:
  • Consider the expression \(x^{\frac{3}{4}} / x^{\frac{1}{4}} = x^{\frac{3}{4} - \frac{1}{4}} = x^{\frac{1}{2}}\).
  • This shows how employing the exponent rule simplifies the division of powers with the same base.
  • It is useful for factoring terms and simplifying expressions.
Using exponent properties makes algebraic manipulation more straightforward, ensuring expressions are both factored and simplified correctly.
Simplifying Expressions
Simplifying expressions is the process of making an algebraic expression as condensed as possible. This involves reducing the expression to its simplest form, where no further factoring or operations are needed.
In our expression \(x^{\frac{1}{4}} (x^{\frac{1}{2}} - 1)\), after factoring out the common factor, simplifying means ensuring that each part of the expression is in its simplest form.
  • The term \(x^{\frac{1}{4}}\) is already simplified as a factor of the entire expression.
  • The expression inside the parentheses, \(x^{\frac{1}{2}} - 1\), is straightforward and cannot be simplified further.
  • Therefore, we have successfully factored and simplified the expression thoroughly.
Simplifying not only makes expressions easier to handle but also highlights their essential structure, making problem-solving more efficient.

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Most popular questions from this chapter

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Will help you prepare for the material covered in the next section. Evaluate $$ \frac{-b-\sqrt{b^{2}-4 a c}}{2 a} $$ for \(a=2, b=9,\) and \(c=-5\)

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