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Chapter 7: Problem 60

For exercises \(55-86\), use prime factorization to find the least common multiple. $$ 12 a^{2} b ; 18 a b^{2} $$

Short Answer

Expert verified
LCM = 36a^2b^2

Step by step solution

01

- Find Prime Factorization of Each Number

First, factorize each of the coefficients (12 and 18) into prime factors.12 = 2^2 * 318 = 2 * 3^2
02

- Include the Variables

Include the variables with their respective powers.12a^2b = 2^2 * 3 * a^2 * b18ab^2 = 2 * 3^2 * a * b^2
03

- Find the Maximum Power of Each Prime Factor

Identify the highest power of each prime factor present in the decompositions:2^2 (from 12), 3^2 (from 18), a^2 (from 12), b^2 (from 18)
04

- Multiply the Maximum Powers Together

Multiply these maximum powers to find the least common multiple (LCM):LCM = 2^2 * 3^2 * a^2 * b^2 = 4 * 9 * a^2 * b^2 = 36a^2b^2

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Key Concepts

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

Prime Factorization
Prime factorization is a method of finding the prime numbers that multiply together to get an original number. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. For example, the prime factors of 12 are 2 and 3, because 2 and 3 are prime, and their product is 12.

In our exercise, we need to find the least common multiple (LCM) of 12 and 18. We start with prime factorization for each number:

12 = 2 脳 2 脳 3, or written with exponents, 12 = 2^2 脳 3
18 = 2 脳 3 脳 3, or 18 = 2 脳 3^2

By breaking the numbers into prime factors, we can more easily compare and combine them to find their LCM.
Variables with Exponents
Next, let's include the variables with their own exponents. Variables in this context are letters like a and b, representing unknown values.

When variables have exponents, it means multiplying the variable by itself a set number of times. For example, \(a^2\) means \(a \times a\).

Let's look at the given expressions: 12a^2b and 18ab^2. These expressions contain both numerical coefficients and variables with exponents.

We express them with their prime factors and variables:

12a^2b = 2^2 脳 3 脳 a^2 脳 b
18ab^2 = 2 脳 3^2 脳 a 脳 b^2
Multiplying Prime Factors
After the prime factors and variables are clear, it's time to find the highest power of each prime factor. This step is crucial for determining the LCM.

From our prime factorizations and variables with exponents, we observe:

2^2 (from 12), 3^2 (from 18), a^2 (from 12), b^2 (from 18)

Each entry in our factorization charts is given with its highest available exponent. We need to take the maximum power of each factor to find the LCM. This ensures that the LCM is divisible by both original numbers.

We compute the LCM by multiplying these maximum powers together:

LCM = 2^2 脳 3^2 脳 a^2 脳 b^2
Algebraic Expressions
An algebraic expression includes numbers, variables, and operators (like plus or minus). In our scenario, the least common multiple can be thought of as an algebraic expression.

Combining all parts, we get:

LCM = 2^2 脳 3^2 脳 a^2 脳 b^2 = 4 脳 9 脳 a^2 脳 b^2 = 36a^2b^2

So, the least common multiple of 12a^2b and 18ab^2 is 36a^2b^2.

Understanding algebraic expressions makes it easier to perform operations like finding the LCM. The key is to break down each component, whether it's a number or a variable raised to a power, and then methodically find and multiply the maximum values for each part.

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Most popular questions from this chapter

The relationship of \(x\) and \(y\) is a direct variation. When \(x=1, y=6\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this direct variation. c. Find \(y\) when \(x=4\), d. Use slope-intercept graphing to graph this equation. e. Use the graph to find \(y\) when \(x=2\).

For a fixed number of hotel rooms, the number of rooms cleaned per hour, \(x\), and the number of hours it takes to clean the rooms, \(y\), is an inverse variation. If a person can clean 8 rooms per hour, it takes 15 hr to clean the rooms. a. Find the constant of variation, \(k\). Include the units of measurement. b. Write an equation that represents this relationship. c. If a person can clean 6 rooms per hour, find the time needed to clean the rooms.

The relationship of the number of tickets sold, \(x\), and the total ticket receipts for an outdoor concert, \(y\), is a direct variation. When 11,000 tickets are sold, the total ticket receipts are \(\$ 495,000\). a. Find the constant of proportionality, \(k\). Include the units of measurement. b. Write an equation that represents this relationship. c. Find the number of tickets sold when the total ticket receipts are \(\$ 562,500\). d. Find the total ticket receipts from the sale of 7575 tickets. e. What does \(k\) represent in this equation?

The relationship of the time a tour guide works, \(x\), and the cost to hire the tour guide, \(y\), is a direct variation. When a tour guide works for \(15 \mathrm{hr}\), the cost is \(\$ 1125\). a. Find the constant of proportionality, \(k\). Include the units of measurement. b. Write an equation that represents this relationship. c. Find the cost to hire a tour guide for \(8 \mathrm{hr}\). d. What does \(k\) represent in this equation?

For exercises \(67-82\), use the five steps and a proportion. In \(2010,3.5\) per 100,000 full-time equivalent workers were killed on the job with a total of 547 workers killed on the job. Find the number of full-time equivalent workers used to create this ratio. Round to the nearest whole number. (Source: www.osha.gov)
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