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Chapter 7: Problem 9

Find the LCD for the fractions in each list. $$ \frac{9}{x^{2}}, \frac{8}{x^{5}} $$

Short Answer

Expert verified
The LCD is \(x^5\).

Step by step solution

01

Identify the Denominators

Look at the denominators of the given fractions. The denominators are: \(x^2\) and \(x^5\).
02

Determine the Highest Power of Each Factor

In order to find the Least Common Denominator (LCD), identify the highest power of each factor from the denominators. Here, the only factor is \(x\). Compare the powers of \(x\) in each denominator: \(x^2\) and \(x^5\).
03

Write the LCD

The LCD is the highest power of all factors present in the denominators. Here, the highest power of \(x\) is \(x^5\). Thus, the LCD is \(x^5\).

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Key Concepts

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

Fractions
Fractions are expressions that represent parts of a whole. They consist of a numerator and a denominator. The numerator is the top part of the fraction and represents how many parts we have. The denominator is the bottom part and indicates the total number of equal parts into which the whole is divided. Understanding fractions is essential when working with algebraic expressions, especially when finding common denominators.

Here are a few key points about fractions:
  • Proper fractions: Numerator is less than the denominator.
  • Improper fractions: Numerator is equal to or greater than the denominator.
  • Mixed numbers: A whole number combined with a proper fraction.
In the context of the given exercise, we are dealing with fractions that have polynomial expressions in their denominators.
Polynomials
Polynomials are algebraic expressions made up of variables and constants combined using addition, subtraction, and multiplication. Each term in a polynomial includes a coefficient and at least one variable raised to a non-negative integer exponent.

Important points about polynomials:
  • A monomial is a polynomial with one term (e.g., 3x or -5).
  • A binomial has two terms (e.g., 4x + 5).
  • A trinomial has three terms (e.g., x^2 + 2x + 1).
Polynomials can sometimes appear in the form of denominators in algebraic fractions, which is the case in our exercise. When dealing with such fractions, it鈥檚 crucial to identify the highest power of the variables to find the Least Common Denominator (LCD).
Exponents
Exponents refer to the power to which a number or variable is raised. They indicate how many times a number (the base) is multiplied by itself. In our exercise, the denominators are polynomials with exponents.

Key points about exponents:
  • The expression x^n means x is multiplied by itself n times (e.g., x^3 = x * x * x).
  • Exponents follow specific arithmetic rules, such as the product of powers (x^a * x^b = x^(a+b)) and the power of a power ( (x^a)^b = x^(a*b) ).
To find the Least Common Denominator in fractions with polynomial denominators, it's essential to determine the highest power of the variable that appears. In the given exercise, since the highest power is x^5, the LCD becomes x^5.

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Most popular questions from this chapter

Add or subtract as indicated. Write each answer in lowest terms. $$ \frac{7}{5}-\frac{3}{4} $$

Solve each variation problem. The area of a circle varies directly as the square of its radius. A circle with radius 3 in. has area 28.278 in. What is the area of a circle with radius 4.1 in. (to the nearest thousandth)?

Rewrite each rational expression with the indicated denominator. $$ \frac{36 r}{r^{2}-r-6}=\frac{?}{(r-3)(r+2)(r+1)} $$

Determine whether each equation represents direct or inverse variation. $$ y=\frac{5}{x} $$

Find each power. $$ (-3)^{2} $$
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