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To find the Least Common Denominator (LCD) of any fractions, you first need to factor the denominators. Factoring means breaking down a number or algebraic expression into its simplest building blocks, or 'factors'. This process makes comparing and combining fractions much easier.

Let's look at our example: We need to find the LCD for the fractions \(\frac{4}{25 m^{3}}\) and \(\frac{7}{10 m^{4}}\). For this, we factor the denominators of each fraction.

For \(25m^3\), the denominator can be broken down as follows:

• 25 is a composite number that factors into 5 and 5, so we have \(5^2\)
• \(m^3\) is already in its prime form as it's a power of a variable

So, the complete factorization of \(25m^3\) is \(5^2 \times m^3\).

Next, for \(10m^4\), the denominator factors as:

• 10 is made up of the prime factors 2 and 5, so we write \(2 \times 5\)
• \(m^4\) is again already in its prime form

The complete factorization of \(10m^4\) is \(2 \times 5 \times m^4\).

Prime factorization involves breaking down a number into its smallest prime factors, which are prime numbers that divide the number exactly without leaving a remainder. This is essential when finding the LCD because it helps identify the common and unique factors in the denominators.

For our exercise, we have:

• \(25 m^3 = 5^2 \times m^3\)
• \(10 m^4 = 2 \times 5 \times m^4\)

The prime factors involved are: 2, 5, and \(m\).

To determine the LCD, we consider all the prime factors present in the denominators of the fractions. This sets the stage for the next step, which involves finding the highest powers of these factors.

Once the denominators are fully factored, the next task is to identify the highest power of each prime factor. The LCD must include the highest power of each factor to cover all the denominators. For our example:

• Factor 2 appears as \(2^1\)
• Factor 5 appears as \(5^2\); note the highest power comes from \(25 m^3\)
• The factor \(m\) appears as \(m^4\); note the highest power from \(10 m^4\)

To find the LCD, we multiply these highest powers together:

\[ \text{LCD} = 2^1 \times 5^2 \times m^4 = 2 \times 25 \times m^4 = 50m^4 \]By including the highest powers of all prime factors present, we ensure that the LCD is the smallest number that each original denominator can divide without leaving a remainder. This common denominator is crucial for adding, subtracting, or comparing fractions effectively.