Problem 45 - Prealgebra and Introductory Algebra

Question

Write each expression in terms of the LCM of the denominators. $$\frac{3}{10^{2}} ; \frac{5}{10^{4}}$$

Step-by-Step Solution

Answer

The expressions in terms of the LCM of the denominators are \(\frac{300}{10^4}\) and \(\frac{5}{10^4}\).

Step 1: Determine the LCM of the Denominators

The given denominators are \(10^2\) and \(10^4\). Largest power of 10 among given denominators is \(10^4\), hence LCM is \(10^4\).

Step 2: Divide the LCM by the first denominator

The first denominator is \(10^2\). Divide the LCM, which is \(10^4\), by \(10^2\) to get \(10^2\).

Step 3: Multiply the first numerator by the result from step 2

The first numerator is 3. Multiply it by \(10^2\) to get \(300\). So, the new fraction in terms of the LCM is \(\frac{300}{10^4}\).

Step 4: Divide the LCM by the second denominator

The second denominator is \(10^4\). Divide the LCM, which is \(10^4\), by \(10^4\) to get 1.

Step 5: Multiply the second numerator by the result from step 4

The second numerator is 5. Multiply it by 1 to get 5. So, the new fraction in terms of the LCM is \(\frac{5}{10^4}\).

Key Concepts

Denominators

When working with fractions, the denominator plays a crucial role. It indicates how many equal parts make up a whole. In the exercise given, the denominators are expressed as powers of 10, namely, \(10^2\) and \(10^4\).

The task is to find their Least Common Multiple (LCM), which is the smallest denominator that can be used to rewrite each fraction without changing their values.

To find the LCM of denominators which are powers of 10, you simply take the highest power. Here, \(10^4\) is the largest power, making it the LCM. This step ensures that both fractions can be expressed with a common denominator, simplifying further calculations or comparisons. Understanding denominators is essential because it affects the entire fraction modification process.

Numerators

The numerators appear at the top part of a fraction and tell us how many parts of the whole we are considering. Working with numerators involves adjusting them according to the new common denominator.

In this exercise, after finding the LCM of the denominators, each numerator is modified. For \(\frac{3}{10^2}\), you multiply the numerator 3 by the result of dividing \(10^4\) by \(10^2\), which is \(10^2\). This gives 300, creating the equivalent fraction \(\frac{300}{10^4}\).

Likewise, for \(\frac{5}{10^4}\), since the denominator already matches the LCM, you multiply 5 by 1, keeping it at \(\frac{5}{10^4}\). Properly handling numerators ensures the fractions remain equivalent even when their expressions change.

Powers of 10

Powers of 10 are a simplified way to represent large numbers and are pivotal in the exercise at hand. When a number is expressed as \(10^n\), it signifies a 1 followed by 'n' zeros. This is very useful in both scientific and everyday mathematical calculations.

The exercise involves denominators \(10^2\) and \(10^4\). Recognizing these as fundamental expressions simplifies the process of finding the LCM and transforming the fractions.

When we say \(10^2\), it equals 100, and \(10^4\) equals 10,000. This knowledge aids in visualizing and conducting both calculations and comparisons without confusion. Powers of 10 allow significant simplification of complex fractions, especially when combining or modifying them.