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Chapter 0: Problem 96

Factor and simplify each algebraic expression. $$12 x^{-\frac{3}{4}}+6 x^{\frac{1}{4}}$$

Short Answer

Expert verified
The simplified form of the given algebraic expression is \(6x^{1/4}(2x^{-1}+1)\)

Step by step solution

01

Find the common factor

Investigate both terms to identify the common factor. In this case, we can observe that both terms share a common factor of \(6x^{1/4}\).
02

Factor out the common factor

Factor out the common factor from both terms. Express each term as the product of the common factor and the rest of the term. Doing so, we get \(6x^{1/4}(2x^{-1}+1)\).
03

Simplify the expression

Simplify the expression inside the bracket as much as possible, according to the properties of exponents. In this case, it simplifies down to \(6x^{1/4}(2x^{-1}+1)\).

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Key Concepts

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

Properties of Exponents
Understanding the properties of exponents is essential when factoring and simplifying algebraic expressions. Exponents indicate how many times a number, known as the base, is multiplied by itself. For example, in the expression \(x^3\), the base is \(x\) and it's used as a factor 3 times: \(x \times x \times x\).

There are several key properties that help with simplification:
  • The Product Rule: \(x^a \times x^b = x^{a+b}\) tells us how to multiply powers with the same base.
  • The Quotient Rule: \(\frac{x^a}{x^b} = x^{a-b}\) guides us when dividing powers with the same base.
  • The Power of a Power Rule: \((x^a)^b = x^{ab}\) explains what to do when a power is raised to another power.
  • The Negative Exponent Rule: \(x^{-a} = \frac{1}{x^a}\) allows us to write negative exponents as fractions.
These properties allow us to manipulate and simplify expressions with exponents more effectively. For instance, if you encounter an expression like \(x^{-\frac{3}{4}}\), understanding that a negative exponent can be written as a reciprocal helps in factoring and simplifying the entire expression.
Simplifying Expressions
Simplifying expressions, put simply, is about making an algebraic expression as straightforward and compact as possible without changing its value. The goal is to combine like terms and use the properties of exponents to condense the expression.

Here's a quick guide to simplifying expressions:
  • Combine like terms: These are terms that have the exact same variables raised to the same powers.
  • Factor out common factors: Just like factoring out the greatest common divisor in numbers, for algebraic expressions, we look for a term that is common to every part of the expression.
  • Apply the properties of exponents: As we've discussed earlier, to simplify exponent-related components within an expression.
When simplifying the given exercise, we look to combine the terms by factoring out the \(6x^{1/4}\), the common factor, which simplifies the expression and makes it cleaner and easier to work with.
Common Factor
Identifying a common factor in algebraic expressions is akin to finding a thread that can unravel a knot. It's the shared element among all terms that can be factored out to simplify the expression. In the context of our exercise, the common factor is \(6x^{1/4}\).

Here's how you can determine the common factor:
  • Review all terms in the expression to see what they share. It might be a number, a variable, or both.
  • Choose the smallest exponent when working with variables, because it's the highest power that can be factored out from each term.
  • The common factor is then used to rewrite each term as a multiple of this factor.
By factoring out \(6x^{1/4}\) from each term, you simplify the overall expression and make subsequent operations, such as addition or multiplication, much easier to handle.

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