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Chapter 11: Problem 29

write the fractions in terms of the LCM of the denominators. $$\frac{4}{x}, \frac{3}{x^{2}}$$

Short Answer

Expert verified
The fractions written in terms of the LCM are \(\frac{4x}{x^{2}}\) and \(\frac{3}{x^{2}}\).

Step by step solution

01

Find the LCM of the denominators

In this case, the denominators are \(x\) and \(x^{2}\). The least common multiple (LCM) of \(x\) and \(x^{2}\) is \(x^{2}\) because \(x^{2}\) contains the highest powers of all variables in \(x\) and \(x^{2}\).
02

Express the fractions in terms of the LCM

Now that we have \(x^{2}\) as our LCM, we need to express each fraction in terms of this LCM. \nFor \(\frac{4}{x}\), we multiply top and bottom by \(x\) to get \(\frac{4x}{x^{2}}\). \nThe fraction \(\frac{3}{x^{2}}\) is already in terms of the LCM, so it remains the same.
03

Final Answer

Now we have the fractions written in terms of the LCM, they are \(\frac{4x}{x^{2}}\) and \(\frac{3}{x^{2}}\) respectively.

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Key Concepts

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

Understanding the Least Common Multiple
When working with fractions, it's often necessary to find a common denominator to perform operations like addition and subtraction. The least common multiple (LCM) of two or more numbers is the smallest number that is a multiple of all the given numbers. For denominators that are algebraic expressions, the process is similar but involves finding the LCM of variables and their exponents.

Finding the LCM in Algebra

When the denominators are algebraic expressions, we look at each variable and its highest power that appears in any of the expressions part of the calculation. For instance, if the denominators are x and x^2, the LCM is x^2, as it includes the variable x raised to the highest power of 2. In doing so, we ensure that we are working with a common base that can be easily manipulated for any algebraic operation involving fractions.
Working with Fractions
Fractions represent parts of a whole and are made up of two parts: the numerator, which is the top number, and the denominator, which is the bottom number. When dealing with fractions that have different denominators, finding a common denominator allows for a direct comparison and combination of these fractions.

Why Use the LCM for Fractions?

The LCM of the denominators simplifies the process of combining fractions. It creates a uniform denominator for all fractions involved in the exercise. By doing this, the numerators can be adjusted accordingly to represent equivalent fractions with a common denominator. Still, the overall value of each fraction remains unchanged. This process is foundational for addition, subtraction, and comparison of fractions within algebraic expressions.
Algebraic Expressions and Fractions
An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operators such as addition or multiplication. When such expressions are present in the denominators of fractions, it can be a bit more complex to manipulate and combine these fractions without a common denominator.

Simplifying Algebraic Fractions

To simplify algebraic expressions in fractions, you must first find a common denominator, just as you would with numerical fractions. After finding the LCM of the algebraic expressions, you adjust each fraction's numerator to match this common base. It is a crucial step because it then allows any further mathematical operations to be performed easily across the fractions, while ensuring that the equations remain balanced and equivalent to their original form.

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Most popular questions from this chapter

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