Problem 52 - Algebra 1: Concepts and Skills

Question

Find the LCD of \(\frac{15}{3 t^{6}}\) and \(\frac{9}{2 t^{4}}\) $$ (A) \frac{1}{6 t^{6}} $$ $$ (B) 6 t^{2} $$ $$ (C) 6 t^{6} $$ $$ (D) 6 t^{10} $$

Step-by-Step Solution

Answer

The LCD of the fractions \(\frac{15}{3 t^{6}}\) and \(\frac{9}{2 t^{4}}\) is \(6 t^{6}\). Hence, the correct option is (C) \(6 t^{6}\).

Step 1 - Identify the Denominators

The denominators of the given fractions are \(3 t^{6}\) and \(2 t^{4}\). These expressions consist of a numerical coefficient and \(t\) terms with different powers.

Step 2 - Find the LCM

The least common multiple (LCM) of the numbers 3 and 2 is 6, which is obtained by multiplying the highest multiple of these numbers. For the \(t\) terms, you take the term with the highest exponent. Here it is \(t^{6}\) because 6 is larger than 4.

Step 3 - Combine the LCM of Number and \(t\) term

Combine 6 (LCM of the numerical coefficients) and \(t^{6}\) (the \(t\) term with the highest exponent) to get the LCD of the given fractions. Therefore, the LCD is \(6 t^{6}\).

Key Concepts

Least Common Multiple (LCM)

When dealing with fractions, especially those that include variables with exponents, finding the Least Common Multiple (LCM) is key to simplifying or adding the fractions. The LCM is the smallest multiple that is exactly divisible by each number in a set. Here's why it's important and how it is determined:

• Identify each number of the given denominators. For instance, if the denominators are 3 and 2, you have these two numbers to consider.
• Next, find the smallest number that each denominator can divide into without leaving a remainder. For 3 and 2, this common number is 6.
• For variables with powers involved, observe that the LCM will use the highest power of the variable present in the set of denominators. This ensures all terms can be expressed using that variable.

Using the LCM accounts for all necessary factors, allowing all fractions concerned to be converted to a common denominator, facilitating operations like addition and subtraction.

Denominators in Fractions

Denominators are the bottom parts of fractions that indicate into how many parts the whole is divided. In problems like finding the LCD of fractions, denominators can include numbers, variables, or even both. Here's how to handle denominators efficiently:

• Recognize the numerical coefficient in each denominator. This helps in setting up the correct sequence for finding common multiples.
• For expressions with variables, such as those with powers like \(t^6\) or \(t^4\), identify these as part of the denominator as well.
• To find the common denominator, you need the LCM of the numerical coefficients and the highest power of each variable present.

Understanding how to decipher and combine these elements leads to an accurate calculation of the least common denominator, making the process of fraction operations simpler and clearer.

Exponents and Powers

Exponents are mathematical notations indicating the number of times a number, known as the base, is multiplied by itself. When simplifying fractions involving variables with exponents, like \(t^6\), understanding their rules is crucial. Here’s why they matter and how to work with them:

• For a given fraction, each variable in the denominator will have an exponent. This represents repeated multiplication.
• When determining the least common multiple among expressions like \(t^6\) and \(t^4\), you focus on the highest exponent. In this case, \(t^6\) is used because it covers the base raised to a greater power over all applicable terms.
• Learning how exponents work helps simplify complex expressions by reducing them to a more workable form.

Working with exponents also means understanding how they interact when combined or compared, like in the pursuit of finding a common denominator for multiple fractions involving such terms.