Least Common Denominators: Videos & Practice Problems

Finding the least common denominator (LCD) for rational expressions involves factoring denominators into prime factors and variables, then selecting the highest powers of each unique factor. For example, the LCD of $\mathrm{30x}$ and $\mathrm{20x}{x}^2$ is $60x^2$. Writing equivalent expressions requires multiplying numerator and denominator by missing factors to match the LCD, ensuring common denominators for addition or subtraction. This process parallels working with rational numbers and polynomials, emphasizing understanding of terms, coefficients, and exponents.

Least Common Denominators of Rational Expressions

Adding or subtracting rational expressions often requires finding a common denominator, especially when the denominators are different. The key to this process is determining the Least Common Denominator (LCD), which is the smallest expression that both denominators can divide into without leaving a remainder. This concept extends from rational numbers to rational expressions, with the main difference being the presence of variables in the denominators.

To find the LCD, start by factoring each denominator completely into its prime factors, including variables. For example, the number 30 factors into 3 × 2 × 5, and 20 factors into 2² × 5. When variables are involved, such as in 30x or 20x², factor the numerical part as before and then include the variable with its exponent (e.g., x or x²).

Once factored, identify the unique prime factors from all denominators. For each prime factor, select the highest power that appears in any denominator. For instance, between 2 and 2², choose 2²; between x and x², choose x². Multiply these highest powers of all unique prime factors together to get the LCD. Using the previous example, the LCD of denominators 30x and 20x² is 3 × 2² × 5 × x², which simplifies to 60x².

When denominators include polynomials, such as quadratics, factor them completely as well. For example, the quadratic expression \(x^2 + 7x + 10\) factors into \((x + 2)(x + 5)\). If another denominator is already factored, like \(x + 5\), include it as is. Then, list all unique factors, considering their highest powers, and multiply them to find the LCD. In this case, the LCD would be \((x + 2)(x + 5)\).

This method ensures that the LCD encompasses all factors necessary to combine rational expressions effectively. Understanding how to factor denominators and select the highest powers of unique prime factors is essential for simplifying and solving problems involving addition or subtraction of rational expressions.

Find the least common denominators of the rational expressions:
$\frac{2y^2+3y-5}{y^2-4}$, $\frac{y-2}{y^2+y-6}$

• A. $\left(y-2\right)\left(y-2\right)\left(y+3\right)$
• B. $y^2-4$
• C. $y^2+y-6$
• D. $\left(y+2\right)\left(y-2\right)\left(y+3\right)$

Answer: (D)

Find the least common denominators of the rational expressions:
$\frac{3x^2-12}{x^2-x-6}$, $\frac{5x+10}{x^2-4x+3}$

• A. $\left(x+2\right)\left(x-3\right)\left(x+1\right)$
• B. $\left(x-1\right)\left(x-3\right)$
• C. $\left(x+2\right)\left(x-3\right)\left(x-1\right)$
• D. $\left(x-2\right)\left(x+3\right)$

Answer: (C)

Find the least common denominators of the rational expressions:
$\frac{2x^2+5x-3}{x^2-9},\frac{4x-12}{x^2-x-6},\frac{x^2+x-2}{x^2-4x+3}$

• A. $\left(x+3\right)\left(x-3\right)\left(x+2\right)\left(x-1\right)$
• B. $\left(x+3\right)\left(x-3\right)^2\left(x+2\right)$
• C. $\left(x-3\right)\left(x-3\right)^2\left(x-2\right)$
• D. $\left(x+3\right)\left(x-3\right)\left(x+2\right)\left(x+1\right)$

Answer: (A)

Write Equivalent Expressions With Common Denominators

Understanding how to write equivalent rational expressions with common denominators builds directly on the concept of finding the least common denominator (LCD). The LCD is essential because it allows us to rewrite rational expressions so they share the same denominator, enabling addition, subtraction, or comparison. This process mirrors the method used with rational numbers, where the LCD is found by breaking down denominators into their prime factors and combining the highest powers of these primes.

For example, when working with rational numbers like \(\frac{1}{30}\) and \(\frac{1}{20}\), the LCD is determined by factoring each denominator: \$30 = 2 \times 3 \times 5\( and \)20 = 2^2 \times 5\(. The LCD is the product of the highest powers of each prime factor, which is \)2^2 \times 3 \times 5 = 60\(. To rewrite each fraction with the denominator 60, identify the missing factors for each denominator relative to the LCD. For 30, the missing factor is 2 (since \)30\( has one 2 but the LCD has \)2^2\(), and for 20, the missing factor is 3. Multiplying numerator and denominator by these missing factors gives equivalent fractions: \)\frac{1}{30} = \frac{2}{60}\( and \)\frac{1}{20} = \frac{3}{60}\(.

This same principle applies to rational expressions involving variables. Consider denominators like \)30x\( and \)20x^2\(. Factoring these gives \)30x = 2 \times 3 \times 5 \times x\( and \)20x^2 = 2^2 \times 5 \times x^2\(. The LCD must include the highest powers of each prime and variable factor, resulting in \)2^2 \times 3 \times 5 \times x^2 = 60x^2\(. To find the missing factors, compare each denominator to the LCD: for \)30x\(, the missing factors are \)2\( and \)x\( (since it has one 2 and one \)x\(, but the LCD has \)2^2\( and \)x^2\(), and for \)20x^2\(, the missing factor is 3.

Rewriting the rational expressions with the LCD as the denominator involves multiplying numerator and denominator by these missing factors. For \)\frac{1}{30x}\(, multiply by \)\frac{2x}{2x}\( to get \)\frac{2x}{60x^2}\(. For \)\frac{1}{20x^2}\(, multiply by \)\frac{3}{3}\( to get \)\frac{3}{60x^2}\(. Both expressions now share the common denominator \)60x^2$, making them equivalent rational expressions.

This method emphasizes the importance of prime factorization and careful comparison of denominators to the LCD. By identifying missing factors and multiplying appropriately, you can rewrite any set of rational expressions with a common denominator, facilitating further operations such as addition or subtraction. Mastery of this process strengthens your ability to manipulate and simplify rational expressions effectively.

Rewrite the expression $\frac{3}{x+2}$ into an equivalent expression having a denominator of $(x+2)(x+5)$.

• A. $\frac{3x+5}{\left(x+2\right)\left(x+5\right)}$
• B. $\frac{3x^2+15}{\left(x+2\right)\left(x+5\right)}$
• C. $\frac{3x+15}{\left(x+2\right)\left(x+5\right)}$
• D. $\frac{3x^2+5}{\left(x+2\right)\left(x+5\right)}$

Answer: (C)

Rewrite the expression $\frac{7x^2+7x}{x^2-1}$ into an equivalent expression having a denominator of $x-1$

• A. $\frac{7x^2}{x-1}$
• B. $\frac{7x^2+7}{x-1}$
• C. $\frac{7x}{x-1}$
• D. $\frac{7x^2-7}{x-1}$

Answer: (C)

Rewrite the expression $\frac{2x^2+2x}{-x^2+1}$ into an equivalent expression having a denominator of $x-1$

• A. $\frac{2x}{x-1}$
• B. $-\frac{2x}{x-1}$
• C. $\frac{2x^2}{x-1}$
• D. $-\frac{2x^2}{x-1}$

Answer: (B)

Write Equivelant Expressions With Common Denominators Example 1

To find the least common denominator (LCD) and rewrite rational expressions with this common denominator, start by identifying all unique prime factors present in each denominator. For example, if the denominators include factors like 3, 7, x, x², and y, list each unique factor, paying special attention to the highest powers of variables. In this case, since both x and x² appear, use the higher power, x², in the LCD.

Once the unique factors are identified, multiply them together to find the LCD. For instance, multiplying 3, 7, x², and y results in the LCD: \$21x^{2}y\(. The goal is to express each rational expression with this denominator.

Next, determine the missing factors for each denominator to reach the LCD. For a denominator like \)3x\(, the missing factors are 7, x, and y. Multiply both the numerator and denominator of this expression by these missing factors to maintain equivalence. This process transforms the expression into one with the denominator \)21x^{2}y\(.

Similarly, for a denominator such as \)7x^{2}y$, identify the missing factor, which in this case is 3. Multiply both numerator and denominator by 3 to achieve the common denominator.

By multiplying numerators and denominators by the appropriate missing factors, both rational expressions become equivalent with the LCD as their denominator. This method ensures that expressions can be easily added, subtracted, or compared by having a consistent denominator.

Understanding how to find the least common denominator and convert rational expressions accordingly is essential for simplifying complex algebraic fractions and solving equations involving rational expressions.

Write Equivelant Expressions With Common Denominators Example 2

When rewriting rational expressions to have a specific denominator, the key is to identify the missing factors needed to match the desired denominator, which acts as the least common denominator (LCD). For example, consider the expression \(\frac{9n}{13}\), and the goal is to rewrite it with the denominator \$13(n - 1)\(. Since the original denominator is 13, the missing factor is \)(n - 1)\(. To maintain equivalence, multiply both the numerator and denominator by this missing factor:

\[\frac{9n}{13} \times \frac{n - 1}{n - 1} = \frac{9n(n - 1)}{13(n - 1)}\]

Distributing in the numerator yields:

\[9n(n - 1) = 9n^2 - 9n\]

and the denominator becomes:

\[13(n - 1) = 13n - 13\]

This results in the equivalent rational expression with the desired denominator:

\[\frac{9n^2 - 9n}{13n - 13}\]

In another example, consider the rational expression \)\frac{10k}{k + 4}\(, and the goal is to rewrite it with the denominator \)k^2 + 9k + 20\(. First, factor the quadratic denominator:

\[k^2 + 9k + 20 = (k + 4)(k + 5)\]

Since the original denominator is \)(k + 4)\(, the missing factor to reach the LCD is \)(k + 5)$. Multiply both numerator and denominator by this missing factor to maintain equivalence:

\[\frac{10k}{k + 4} \times \frac{k + 5}{k + 5} = \frac{10k(k + 5)}{(k + 4)(k + 5)}\]

Distribute in the numerator:

\[10k(k + 5) = 10k^2 + 50k\]

The denominator is the factored form of the quadratic:

\[ (k + 4)(k + 5) = k^2 + 9k + 20 \]

This gives the equivalent rational expression:

\[\frac{10k^2 + 50k}{k^2 + 9k + 20}\]

By identifying missing factors and multiplying numerator and denominator accordingly, you can rewrite rational expressions with any desired denominator, ensuring equivalence throughout the process. Factoring polynomials and understanding the structure of denominators are essential skills for manipulating rational expressions effectively.

Here’s what students ask on this topic:

What is the least common denominator (LCD) and how do you find it for rational expressions?

The least common denominator (LCD) is the smallest expression that can serve as a common denominator for two or more rational expressions. To find the LCD for rational expressions, first factor each denominator completely into prime factors and variables with their exponents. Then, identify all unique factors and select the highest power of each factor present in any denominator. Finally, multiply these highest powers together to get the LCD. For example, for denominators $\mathrm{30x}$ and ${\mathrm{20x}}^2$, factor them as $2\cdot 3\cdot 5\cdot x$ and $2^2\cdot 5\cdot x^2$. The LCD is $2{\cdot }^2\cdot 3\cdot 5\cdot x^2$, which equals $\mathrm{60x}{x}^2$. This process ensures you have the smallest common denominator for adding or subtracting rational expressions.

How do you write equivalent rational expressions with a common denominator?

To write equivalent rational expressions with a common denominator, first find the least common denominator (LCD) of the given expressions. Then, determine the missing factors in each denominator compared to the LCD. Multiply both the numerator and denominator of each rational expression by these missing factors to create equivalent expressions with the LCD as the denominator. For example, if the LCD is $\mathrm{60x}{x}^2$ and one denominator is $\mathrm{30x}$, the missing factors are $2\cdot x$. Multiply numerator and denominator by $\mathrm{2x}$ to get an equivalent expression with denominator $\mathrm{60x}{x}^2$. This method ensures all expressions share the same denominator, allowing for addition or subtraction.

Why is factoring denominators important when finding the least common denominator?

Factoring denominators is crucial because it breaks down complex expressions into their prime factors and variables, making it easier to identify unique factors and their highest powers. This step ensures the least common denominator (LCD) includes all necessary factors to accommodate each denominator. Without factoring, you might miss factors or powers, leading to an incorrect or non-minimal common denominator. For example, factoring $\mathrm{20x}{x}^2$ into $2^2\cdot 5\cdot x^2$ reveals the powers of 2 and x, which are essential for determining the LCD accurately.

How do you handle variables with exponents when finding the LCD of rational expressions?

When variables with exponents appear in denominators, treat them like prime factors. Identify the highest exponent of each variable across all denominators. The LCD must include each variable raised to its highest power. For example, if one denominator has $x$ and another has $x^2$, the LCD includes $x^2$ because 2 is the highest exponent. This ensures the LCD can be divided evenly by each denominator, allowing for proper addition or subtraction of rational expressions.

Can you explain the process of finding the LCD for rational expressions with quadratic denominators?

For quadratic denominators, first factor the quadratic expression completely into linear factors if possible. For example, factor $x^2+\; 7x\; +\; 10$ into $(x\; +\; 2)(x\; +\; 5)$. Then, list all unique factors from all denominators, including these linear factors. The LCD is the product of these unique factors, each taken at their highest power. This approach extends the prime factorization method to polynomials, ensuring the LCD accommodates all denominators for addition or subtraction.