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Chapter 6: Problem 10

Find the LCD for the fractions in each list. $$ \frac{12}{m^{7}}, \frac{14}{m^{8}} $$

Short Answer

Expert verified
The LCD is \(m^8\).

Step by step solution

01

Identify the Denominators

The denominators of the given fractions are \(m^7\) and \(m^8\).
02

Determine the Least Common Denominator (LCD)

To find the least common denominator (LCD) of \(m^7\) and \(m^8\), we need to find the highest power of the variable \(m\) that appears in each denominator. In this case, the higher power is \(m^8\).
03

Verify the LCD

Check that \(m^8\) is divisible by both \(m^7\) and \(m^8\). Since \(m^8\) contains \(m^7\) as a factor and is itself, \(m^8\) is indeed the least common denominator.

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Key Concepts

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

Fractions
Fractions are mathematical expressions that represent a part of a whole. They consist of a numerator (the top part) and a denominator (the bottom part). The numerator indicates how many parts we have, while the denominator tells us into how many equal parts the whole is divided.
For example, in the fraction \( \frac{12}{m^{7}} \), the numerator is 12, and the denominator is \(m^{7}\). Fractions can represent rational numbers, ratios, or even parts of algebraic expressions.
  • Fractions are essential for solving various math problems.
  • They help in comparing sizes, measuring quantities, and performing calculations.
Understanding fractions deeply can help make math easier and more intuitive.
Denominators
The denominator in a fraction is the number below the line (or in this case, the expression), which shows into how many parts the whole is divided.
For the fractions in the given exercise, the denominators are \(m^7\) and \(m^8\). When comparing or adding fractions, it's crucial to have a common denominator.
The Least Common Denominator (LCD) is the smallest denominator that can be used for all fractions involved, ensuring they can be easily compared or added. To find the LCD, we need to look at:
  • The highest powers of any variables present in the denominators.
This helps us identify the smallest shared multiple.
Highest Power of Variables
In algebraic fractions, denominators may contain variables raised to different powers.
In our exercise, the denominators are \(m^7\) and \(m^8\). The key to finding the Least Common Denominator is identifying the highest power of these variables.
Variables in the denominators can complicate things, but there's a straightforward method to handle them:
  • Look at each variable and its highest power across all denominators.
  • In our case, between \(m^7\) and \(m^8\), the higher power is \(m^8\).
Thus, \(m^8\) is our Least Common Denominator. This process ensures that the LCD can hold all parts of the original fractions, allowing for easy addition or comparison.

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Most popular questions from this chapter

Which one of these rational expressions is not equivalent to \(\frac{x-3}{4-x} ?\) A. \(\frac{3-x}{x-4}\) B. \(\frac{x+3}{4+x}\) c. \(-\frac{3-x}{4-x}\) D. \(-\frac{x-3}{x-4}\)

Multiply or divide. Write each answer in lowest terms. $$ \frac{2 k^{2}+3 k-2}{6 k^{2}-7 k+2} \cdot \frac{4 k^{2}-5 k+1}{k^{2}+k-2} $$

involve factoring by grouping (Section 5.1) and factoring sums and differences of cubes (Section 5.4). Write each rational expression in lowest terms $$ \frac{x^{3}-27}{x-3} $$

involve factoring by grouping (Section 5.1) and factoring sums and differences of cubes (Section 5.4). Write each rational expression in lowest terms $$ \frac{x^{2} y+y+x^{2} z+z}{x y+x z} $$

Only one of these choices is equal to \(\frac{\frac{1}{3}+\frac{1}{12}}{\frac{1}{2}+\frac{1}{4}} .\) Which one is it? Answer this question without showing any work, and explain your reasoning. A. \(\frac{5}{9}\) B. \(-\frac{5}{9}\) C. \(-\frac{9}{5}\) D. \(-\frac{1}{12}\)
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