Find The Least Common Denominator Of $\frac{1}{5x-25}$ And $\frac{3x}{4x-20}$.

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Introduction

In mathematics, the least common denominator (LCD) is an essential concept in algebra, particularly when dealing with rational expressions. It is the smallest multiple that two or more denominators have in common. In this article, we will focus on finding the least common denominator of two rational expressions: 15xโˆ’25\frac{1}{5x-25} and 3x4xโˆ’20\frac{3x}{4x-20}.

Understanding Rational Expressions

A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. Rational expressions can be simplified, added, subtracted, multiplied, and divided, just like regular fractions. However, when dealing with rational expressions, we need to consider the restrictions on the variables, which are the values that make the denominator equal to zero.

Factoring the Denominators

To find the least common denominator, we need to factor the denominators of both rational expressions. The first rational expression is 15xโˆ’25\frac{1}{5x-25}, and the second rational expression is 3x4xโˆ’20\frac{3x}{4x-20}. We can factor the denominators as follows:

  • 5xโˆ’25=5(xโˆ’5)5x-25 = 5(x-5)
  • 4xโˆ’20=4(xโˆ’5)4x-20 = 4(x-5)

Finding the Least Common Denominator

Now that we have factored the denominators, we can find the least common denominator. The least common denominator is the product of the highest power of each unique factor. In this case, the unique factors are 55 and 44, and the highest power of each factor is 11. Therefore, the least common denominator is:

5(xโˆ’5)โ‹…4(xโˆ’5)=20(xโˆ’5)25(x-5) \cdot 4(x-5) = 20(x-5)^2

Simplifying the Rational Expressions

Now that we have found the least common denominator, we can simplify the rational expressions by multiplying each expression by the necessary factors to obtain the least common denominator. We can do this by multiplying the numerator and denominator of each expression by the necessary factors.

  • 15xโˆ’25=1โ‹…4(xโˆ’5)5(xโˆ’5)โ‹…4(xโˆ’5)=4(xโˆ’5)20(xโˆ’5)2\frac{1}{5x-25} = \frac{1 \cdot 4(x-5)}{5(x-5) \cdot 4(x-5)} = \frac{4(x-5)}{20(x-5)^2}
  • 3x4xโˆ’20=3xโ‹…5(xโˆ’5)4(xโˆ’5)โ‹…5(xโˆ’5)=15x(xโˆ’5)20(xโˆ’5)2\frac{3x}{4x-20} = \frac{3x \cdot 5(x-5)}{4(x-5) \cdot 5(x-5)} = \frac{15x(x-5)}{20(x-5)^2}

Combining the Rational Expressions

Now that we have simplified the rational expressions, we can combine them by adding or subtracting the numerators. In this case, we will add the numerators:

4(xโˆ’5)20(xโˆ’5)2+15x(xโˆ’5)20(xโˆ’5)2=4(xโˆ’5)+15x(xโˆ’5)20(xโˆ’5)2\frac{4(x-5)}{20(x-5)^2} + \frac{15x(x-5)}{20(x-5)^2} = \frac{4(x-5) + 15x(x-5)}{20(x-5)^2}

Simplifying the Result

We can simplify the result by combining like terms in the numerator:

4(xโˆ’5)+15x(xโˆ’5)20(xโˆ’5)2=(4+15x)(xโˆ’5)20(xโˆ’5)2\frac{4(x-5) + 15x(x-5)}{20(x-5)^2} = \frac{(4+15x)(x-5)}{20(x-5)^2}

Conclusion

In conclusion, finding the least common denominator of two rational expressions involves factoring the denominators, finding the product of the highest power of each unique factor, and simplifying the rational expressions by multiplying each expression by the necessary factors to obtain the least common denominator. By following these steps, we can combine the rational expressions and simplify the result.

Final Answer

The final answer is (4+15x)(xโˆ’5)20(xโˆ’5)2\boxed{\frac{(4+15x)(x-5)}{20(x-5)^2}}.

References

  • [1] Algebra, 2nd ed. by Michael Artin
  • [2] Calculus, 3rd ed. by Michael Spivak

Additional Resources

  • Khan Academy: Rational Expressions
  • MIT OpenCourseWare: Algebra
  • Wolfram Alpha: Rational Expressions

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Q: What is the least common denominator (LCD)?

A: The least common denominator is the smallest multiple that two or more denominators have in common. It is an essential concept in algebra, particularly when dealing with rational expressions.

Q: Why is finding the LCD important?

A: Finding the LCD is important because it allows us to combine rational expressions by adding or subtracting the numerators. This is a crucial step in simplifying and solving rational expressions.

Q: How do I find the LCD of two rational expressions?

A: To find the LCD of two rational expressions, you need to factor the denominators, find the product of the highest power of each unique factor, and simplify the rational expressions by multiplying each expression by the necessary factors to obtain the LCD.

Q: What if the denominators have different variables?

A: If the denominators have different variables, you need to find the least common multiple (LCM) of the variables. The LCM is the smallest multiple that two or more variables have in common.

Q: How do I find the LCM of two variables?

A: To find the LCM of two variables, you need to list the multiples of each variable and find the smallest multiple that appears in both lists.

Q: What if the denominators have the same variable but different exponents?

A: If the denominators have the same variable but different exponents, you need to find the highest power of the variable that appears in both denominators.

Q: Can I use a calculator to find the LCD?

A: Yes, you can use a calculator to find the LCD. However, it's essential to understand the concept and be able to apply it manually.

Q: What are some common mistakes to avoid when finding the LCD?

A: Some common mistakes to avoid when finding the LCD include:

  • Not factoring the denominators correctly
  • Not finding the product of the highest power of each unique factor
  • Not simplifying the rational expressions correctly
  • Not combining the rational expressions correctly

Q: How do I check my answer for the LCD?

A: To check your answer for the LCD, you can multiply the numerator and denominator of each rational expression by the LCD and simplify the result. If the result is the same as the original expression, then your answer is correct.

Q: What are some real-world applications of finding the LCD?

A: Finding the LCD has many real-world applications, including:

  • Simplifying and solving rational expressions in algebra
  • Finding the area and perimeter of shapes with rational dimensions
  • Solving problems in physics and engineering that involve rational expressions

Q: Can I use the LCD to solve equations with rational expressions?

A: Yes, you can use the LCD to solve equations with rational expressions. By multiplying both sides of the equation by the LCD, you can eliminate the denominators and solve for the variable.

Q: What are some tips for finding the LCD quickly and accurately?

A: Some tips for finding the LCD quickly and accurately include:

  • Factoring the denominators correctly
  • Finding the product of the highest power of each unique factor
  • Simplifying the rational expressions correctly
  • Combining the rational expressions correctly
  • Checking your answer carefully

Q: Can I use technology to find the LCD?

A: Yes, you can use technology to find the LCD. Many calculators and computer algebra systems can find the LCD quickly and accurately.

Q: What are some common mistakes to avoid when using technology to find the LCD?

A: Some common mistakes to avoid when using technology to find the LCD include:

  • Not entering the correct input
  • Not selecting the correct function
  • Not checking the answer carefully
  • Not understanding the concept of the LCD

Q: How do I know if I need to find the LCD?

A: You need to find the LCD when you are working with rational expressions and need to combine them by adding or subtracting the numerators.

Q: What are some real-world examples of finding the LCD?

A: Some real-world examples of finding the LCD include:

  • Simplifying and solving rational expressions in algebra
  • Finding the area and perimeter of shapes with rational dimensions
  • Solving problems in physics and engineering that involve rational expressions

Q: Can I use the LCD to solve problems in other areas of mathematics?

A: Yes, you can use the LCD to solve problems in other areas of mathematics, including geometry, trigonometry, and calculus.

Q: What are some tips for finding the LCD in other areas of mathematics?

A: Some tips for finding the LCD in other areas of mathematics include:

  • Understanding the concept of the LCD
  • Factoring the denominators correctly
  • Finding the product of the highest power of each unique factor
  • Simplifying the rational expressions correctly
  • Combining the rational expressions correctly
  • Checking your answer carefully

Q: Can I use the LCD to solve problems in science and engineering?

A: Yes, you can use the LCD to solve problems in science and engineering, including physics and engineering.

Q: What are some real-world examples of finding the LCD in science and engineering?

A: Some real-world examples of finding the LCD in science and engineering include:

  • Simplifying and solving rational expressions in physics and engineering
  • Finding the area and perimeter of shapes with rational dimensions
  • Solving problems in physics and engineering that involve rational expressions

Q: Can I use the LCD to solve problems in finance and economics?

A: Yes, you can use the LCD to solve problems in finance and economics, including calculating interest rates and investment returns.

Q: What are some real-world examples of finding the LCD in finance and economics?

A: Some real-world examples of finding the LCD in finance and economics include:

  • Calculating interest rates and investment returns
  • Finding the area and perimeter of shapes with rational dimensions
  • Solving problems in finance and economics that involve rational expressions

Q: Can I use the LCD to solve problems in other areas of science and engineering?

A: Yes, you can use the LCD to solve problems in other areas of science and engineering, including chemistry and biology.

Q: What are some real-world examples of finding the LCD in other areas of science and engineering?

A: Some real-world examples of finding the LCD in other areas of science and engineering include:

  • Simplifying and solving rational expressions in chemistry and biology
  • Finding the area and perimeter of shapes with rational dimensions
  • Solving problems in chemistry and biology that involve rational expressions