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 $\mathrm{20x^2}$ is $60x^2$. Writing equivalent expressions requires multiplying numerator and denominator by missing factors to match the LCD, enabling addition or subtraction of rational expressions. This process parallels working with rational numbers and is essential for simplifying multivariable polynomials and understanding terms, coefficients, and exponents in algebraic expressions.

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 divide into evenly. This concept is closely related to finding the least common denominator for rational numbers, where the LCD is the product of the unique prime factors raised to their highest powers.

To find the LCD, start by factoring each denominator completely into its prime factors. For example, the number 30 factors into 3 × 2 × 5, and 20 factors into 2² × 5. When listing unique prime factors, include each prime factor only once, using the highest exponent found among the denominators. In this case, the unique prime factors are 3, 2², and 5, so the LCD is calculated as \$3 \times 2^{2} \times 5 = 60\(.

This method extends naturally to rational expressions that include variables. For instance, consider denominators like 30x and 20x². Factor the numerical part as before, then include the variable factors with their exponents. Here, 30x factors as \)3 \times 2 \times 5 \times x\(, and 20x² factors as \)2^{2} \times 5 \times x^{2}\(. The LCD includes the highest powers of each prime and variable factor, resulting in \)3 \times 2^{2} \times 5 \times x^{2} = 60x^{2}\(.

When denominators involve polynomials, such as \)x + 5\( or quadratic expressions like \)x^{2} + 7x + 10\(, factor them completely. The quadratic \)x^{2} + 7x + 10\( factors into \)(x + 2)(x + 5)\( because 2 and 5 multiply to 10 and add to 7. The LCD is then the product of all unique factors, considering their highest powers. For example, with denominators \)2x(x + 5)\( and \)x^{2} + 7x + 10\(, the LCD is \)2x(x + 2)(x + 5)\(, but since \)2x\( is already factored, and \)x + 5\( appears in both, the unique factors are \)2\(, \)x\(, \)x + 2\(, and \)x + 5$.

In summary, finding the least common denominator involves fully factoring each denominator, identifying all unique prime and polynomial factors, and then multiplying these factors together using the highest powers present. This approach ensures that rational expressions can be added or subtracted by rewriting each with the same denominator, facilitating further simplification or solution of equations.

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

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. $(x+2)(x-3)(x+1)$
• B. $(x-1)(x-3)$
• C. $(x+2)(x-3)(x-1)$
• D. $(x-2)(x+3)$

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. $(x+3)(x-3)(x+2)(x-1)$
• B. $(x+3){(x-3)}^2(x+2)$
• C. $(x-3){(x-3)}^2(x-2)$
• D. $(x+3)(x-3)(x+2)(x+1)$

Answer: (A)

Write Equivelant 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). This process is essential for simplifying, adding, or subtracting rational expressions. The key is to first determine the LCD by breaking down each denominator into its prime factors and variables, then identifying the highest powers of each factor present.

For example, when working with rational numbers such as fractions with denominators 30 and 20, the LCD is found by taking the prime factorization: \$30 = 2 \times 3 \times 5\( and \)20 = 2^2 \times 5\(. The LCD must include the highest powers of each prime factor, which results in \)2^2 \times 3 \times 5 = 60\(. To rewrite each fraction with this common denominator, identify the missing factors in each denominator compared to the LCD. For 30, the missing factor is \)2\( (since 30 has one 2 but the LCD has \)2^2\(), and for 20, the missing factor is \)3\(. Multiplying the numerator and denominator of each fraction by these missing factors creates equivalent fractions with denominator 60, such as \)\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\(. The LCD here includes the prime factors and the highest powers of variables: \)2^2 \times 3 \times 5 \times x^2 = 60x^2\(. To find the missing factors, compare each denominator to the LCD. For \)30x = 2 \times 3 \times 5 \times x\(, the missing factors are \)2\( and \)x\( to reach \)2^2\( and \)x^2\(. For \)20x^2 = 2^2 \times 5 \times x^2\(, the missing factor is \)3\(. Multiplying the numerator and denominator of each rational expression by these missing factors yields equivalent expressions with the common denominator \)60x^2\(:

\)\frac{1}{30x} = \frac{2x}{60x^2}\( and \)\frac{1}{20x^2} = \frac{3}{60x^2}$.

This method ensures that all rational expressions share the same denominator, facilitating operations like addition or subtraction. The process hinges on prime factorization and careful comparison to the LCD, emphasizing the importance of recognizing missing factors and multiplying appropriately to maintain equivalence.

Mastering how to write equivalent rational expressions with common denominators is a foundational skill in algebra that simplifies working with complex fractions and rational expressions. Practice with various examples strengthens understanding and fluency in this essential algebraic technique.

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

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

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 form 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 in each denominator relative to the LCD. For a denominator like \)3x\(, the missing factors to reach \)21x^{2}y\( are 7, x, and y. Multiply both the numerator and denominator of this rational expression by these missing factors to maintain equivalence. This process ensures the denominator becomes \)21x^{2}y\(.

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

By applying these steps, each rational expression is rewritten with the least common denominator, allowing for straightforward addition, subtraction, or comparison. This method emphasizes understanding prime factorization, recognizing the highest powers of variables, and maintaining equivalence through multiplication by one (expressed as the missing factors over themselves).

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 gives:

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

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)\(, the missing factor is \)(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, which can be expanded back to:

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

This process ensures the rational expression is equivalent but now has the desired denominator. Understanding how to factor polynomials and identify missing factors is essential for rewriting rational expressions with specific denominators. This technique is fundamental in simplifying complex rational expressions and preparing them for addition, subtraction, or comparison.

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 $\mathrm{20x}{x}^2$, factor them as $3imes2imes5imesx$ and $2^{2}imes5imes{x}^2$. The LCD is $3imes{2}^{2}imes5imes{x}^2=\mathrm{60x}{x}^2$. 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 new denominator. For example, if the LCD is $\mathrm{60x}{x}^2$ and one denominator is $\mathrm{30x}$, the missing factors are $2imesx$. Multiply numerator and denominator by $\mathrm{2x}$ to get an equivalent expression with denominator $\mathrm{60x}{x}^2$. This process allows you to add or subtract rational expressions by having a common denominator.

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

Factoring denominators is crucial when finding the least common denominator (LCD) because it breaks down each denominator into its prime factors and variables with exponents. This detailed breakdown allows you to identify all unique factors and their highest powers across the denominators. Without factoring, you might miss common factors or incorrectly determine the LCD, leading to errors in simplifying or combining rational expressions. Factoring ensures accuracy by revealing the building blocks of each denominator, making it easier to select the correct factors to multiply for the LCD. This method parallels finding the LCD for rational numbers but extends to variables and exponents in algebraic expressions.

How do you handle variables with exponents when finding the least common denominator?

When finding the least common denominator (LCD) involving variables with exponents, treat the variables like prime factors. Identify the variable factors in each denominator and select the highest exponent of each variable present. For example, if one denominator has $x$ and another has $x^2$, the LCD will include $x^2$ because it is the highest power. This ensures the LCD can accommodate all denominators. Multiplying by the missing variable powers when rewriting equivalent expressions guarantees the denominators match the LCD, allowing for proper addition or subtraction of rational expressions.

Can you explain the process of finding the LCD for rational expressions with an example?

Certainly! To find the LCD for rational expressions, start by factoring each denominator completely. For example, consider the denominators $\mathrm{30x}$ and $x^2+\; 7x+\; 10$. Factor $30x$ as $3\; imes\; 2\; imes\; 5\; imes\; x$. The quadratic $x^2\; +\; 7x\; +\; 10$ factors into $(x\; +\; 2)(x\; +\; 5)$. Now list all unique factors: $3,\; 2,\; 5,\; x,\; (x\; +\; 2),\; (x\; +\; 5)$. The LCD is the product of these unique factors: $3\; imes\; 2\; imes\; 5\; imes\; x\; imes\; (x\; +\; 2)\; imes\; (x\; +\; 5)$. This LCD can be used to rewrite rational expressions with these denominators to have a common denominator for addition or subtraction.