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Chapter 0: Problem 95

Factor and simplify each algebraic expression. $$4 x^{-\frac{2}{3}}+8 x^{\frac{1}{3}}$$

Short Answer

Expert verified
The factored and simplified form of the expression is \( 4x^{-\frac{2}{3}}(1+2x) \).

Step by step solution

01

Identify common factors

Looking at each term, we can see that \( x^{-\frac{2}{3}} \) is a factor of both terms in the given expression, and so is 4. Therefore, we can factor out \( 4x^{-\frac{2}{3}} \).
02

Factor out the common factors

We proceed to factor out the common factors from the expression, giving \( 4x^{-\frac{2}{3}}(1+2x)\).
03

Simplify the expression

The expression is already simplified since there is no further simplification that can be done to \( 4x^{-\frac{2}{3}}(1+2x) \).

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Key Concepts

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

Factoring in Algebraic Expressions
Factoring is a fundamental process in algebra that involves identifying components or "factors" within an expression that can be taken out of each term to simplify it. In algebraic terms, we often look for the greatest common factor among terms.
When we examined the expression \(4x^{-\frac{2}{3}} + 8x^{\frac{1}{3}}\), the number 4 and the expression \(x^{-\frac{2}{3}}\) were common factors.
This means both terms in the expression could be divided by them. By factoring these out, we simplify the expression to \(4x^{-\frac{2}{3}}(1 + 2x)\).
Factoring makes expressions easier to work with, especially when solving equations or reducing the complexity of algebraic manipulations.
Simplification of Algebraic Expressions
Simplification is the process of making an algebraic expression as concise as possible. This involves reducing the number of terms, getting rid of complex fractions, or simplifying complex operations.
In the given problem, after factoring, we are left with the expression \(4x^{-\frac{2}{3}}(1+2x)\). There are no further like terms to combine, and no fractions or nested operations to simplify.
  • Check for any possible simplifications, like combining like terms.
  • Evaluate exponents carefully to ensure all terms are in their simplest form.
In this case, the expression is already in its simplest form after the factoring process.
Understanding Exponents in Algebra
Exponents are shorthand for expressing repeated multiplication. They are crucial in algebra for representing large numbers or complex expressions succinctly.
In our problem, we encounter negative and fractional exponents, such as \(x^{-\frac{2}{3}}\) and \(x^{\frac{1}{3}}\).
Fractional exponents indicate roots 鈥� for example, \(x^{\frac{1}{3}}\) is the cube root of \(x\), and negative exponents indicate reciprocals 鈥� \(x^{-\frac{2}{3}}\) is the reciprocal of \(x^{\frac{2}{3}}\).
  • Fractional exponents are another way of expressing roots.
  • Negative exponents indicate that the term is on the denominator of a fraction.
Understanding how to work with exponents allows us to simplify expressions accurately and to factor them effectively, which is vital for algebraic problem-solving.

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Most popular questions from this chapter

The bar graph shows the percentage of U.S. college freshmen with an average grade of A in high school. (GRAPH CAN NOT COPY) The data displayed by the bar graph can be described by the mathematical model $$p=\frac{4 x}{5}+25$$ where \(x\) is the number of years after 1980 and \(p\) is the percentage of U.S. college freshmen who had an average grade of A in high school. Use this information a. According to the formula, in 2010 , what percentage of U.S. college freshmen had an average grade of \(A\) in high school? Does this underestimate or overestimate the percent displayed by the bar graph? By how much? b. If trends shown by the formula continue, project when \(57 \%\) of U.S. college freshmen will have had an average grade of A in high school.

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Will help you prepare for the material covered in the first section of the next chapter. If \(y=4-x,\) find the value of \(y\) that corresponds to values of \(x\) for each integer starting with \(-3\) and ending with 3

What does it mean to solve a formula for a variable?
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