Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
#4
![### Finding the Lowest Common Denominator (LCD)
In this exercise, we aim to find the lowest common denominator for the given fractions.
The fractions are as follows:
\[
\frac{3a^2}{a^2 + a + 1}, \quad \frac{2a}{a - 1}, \quad \frac{a^3}{a^3 - 1}
\]
The options for the lowest common denominator are:
- \((a^3 - 1)(a^2 + a + 1)\)
- \((a + 1)^2(a - 1)\)
- \(a^3 - 1\)
To find the LCD, we must first factorize the denominators (if possible) and then determine the least common multiple.
1. For the fraction \(\frac{3a^2}{a^2 + a + 1}\), the denominator \(a^2 + a + 1\) cannot be further factored using real numbers.
2. For the fraction \(\frac{2a}{a - 1}\), the denominator \(a - 1\) is already in its simplest form.
3. For the fraction \(\frac{a^3}{a^3 - 1}\), we can factorize \(a^3 - 1\) as follows:
\[
a^3 - 1 = (a - 1)(a^2 + a + 1)
\]
Given these factors, the least common multiple of the denominators must include each distinct factor the maximum number of times it appears in any factorization. Here, the common denominator is the product of \((a - 1)\) and \((a^2 + a + 1)\):
Hence, the lowest common denominator is:
\[
(a^3 - 1)(a^2 + a + 1)
\]
Thus, the correct option is:
- \((a^3 - 1)(a^2 + a + 1)\)
This concludes the analysis for finding the lowest common denominator for the given fractions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5f7b3938-790b-4169-a858-04369df79473%2Feb2c74b5-44a6-4eb8-9e9b-a378a2236f82%2F292lk38_processed.png&w=3840&q=75)
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