Combine The Following Fractions And Express The Result In Fully Reduced Form. { \frac{7}{6x} + \frac{1}{6x^2}$}$

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Introduction

In mathematics, combining fractions with different denominators is a fundamental concept that requires a thorough understanding of algebraic manipulations. When dealing with fractions, it is essential to find a common denominator to add or subtract them. In this article, we will explore how to combine the given fractions and express the result in fully reduced form.

Understanding the Problem

The given problem involves combining two fractions with different denominators:

76x+16x2\frac{7}{6x} + \frac{1}{6x^2}

To combine these fractions, we need to find a common denominator. The least common multiple (LCM) of the denominators 6x6x and 6x26x^2 is 6x26x^2. Therefore, we will multiply the first fraction by xx\frac{x}{x} to make its denominator 6x26x^2.

Step 1: Multiply the First Fraction

To make the denominator of the first fraction 6x26x^2, we will multiply it by xx\frac{x}{x}.

76x⋅xx=7x6x2\frac{7}{6x} \cdot \frac{x}{x} = \frac{7x}{6x^2}

Step 2: Add the Fractions

Now that both fractions have the same denominator, we can add them.

7x6x2+16x2=7x+16x2\frac{7x}{6x^2} + \frac{1}{6x^2} = \frac{7x + 1}{6x^2}

Simplifying the Result

The resulting fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD). In this case, the GCD of 7x+17x + 1 and 6x26x^2 is 1. Therefore, the fraction is already in its simplest form.

Conclusion

Combining fractions with different denominators requires finding a common denominator and then adding or subtracting the fractions. In this article, we have demonstrated how to combine the given fractions and express the result in fully reduced form. By following these steps, you can simplify complex fractions and solve various mathematical problems.

Common Denominators and Least Common Multiples

When dealing with fractions, it is essential to understand the concept of common denominators and least common multiples (LCMs). A common denominator is the smallest multiple that two or more denominators have in common. The LCM of two or more numbers is the smallest number that is a multiple of each of the given numbers.

Example 1: Finding a Common Denominator

Suppose we want to add the fractions 14\frac{1}{4} and 16\frac{1}{6}. To find a common denominator, we need to find the LCM of 4 and 6. The LCM of 4 and 6 is 12. Therefore, we can rewrite the fractions as follows:

14=312\frac{1}{4} = \frac{3}{12} and 16=212\frac{1}{6} = \frac{2}{12}

Now that both fractions have the same denominator, we can add them.

312+212=512\frac{3}{12} + \frac{2}{12} = \frac{5}{12}

Example 2: Finding the Least Common Multiple

Suppose we want to find the LCM of 8 and 12. To do this, we need to list the multiples of each number and find the smallest number that appears in both lists.

Multiples of 8: 8, 16, 24, 32, ... Multiples of 12: 12, 24, 36, 48, ...

The smallest number that appears in both lists is 24. Therefore, the LCM of 8 and 12 is 24.

Real-World Applications

Combining fractions with different denominators has numerous real-world applications in various fields, including science, engineering, and finance. For example, in physics, combining fractions is used to calculate the acceleration of an object. In engineering, combining fractions is used to calculate the stress on a material. In finance, combining fractions is used to calculate the interest rate on a loan.

Conclusion

Introduction

In our previous article, we explored how to combine fractions with different denominators and express the result in fully reduced form. In this article, we will answer some frequently asked questions (FAQs) related to combining fractions with different denominators.

Q: What is the least common multiple (LCM) of two or more numbers?

A: The LCM of two or more numbers is the smallest number that is a multiple of each of the given numbers. For example, the LCM of 4 and 6 is 12.

Q: How do I find the LCM of two or more numbers?

A: To find the LCM of two or more numbers, you can list the multiples of each number and find the smallest number that appears in both lists. Alternatively, you can use the following formula:

LCM(a, b) = (a × b) / GCD(a, b)

where GCD(a, b) is the greatest common divisor of a and b.

Q: What is the greatest common divisor (GCD) of two or more numbers?

A: The GCD of two or more numbers is the largest number that divides each of the given numbers without leaving a remainder. For example, the GCD of 12 and 18 is 6.

Q: How do I find the GCD of two or more numbers?

A: To find the GCD of two or more numbers, you can use the following methods:

  1. List the factors of each number and find the largest number that appears in both lists.
  2. Use the Euclidean algorithm to find the GCD.
  3. Use the following formula:

GCD(a, b) = (a × b) / LCM(a, b)

Q: Can I add or subtract fractions with different denominators?

A: Yes, you can add or subtract fractions with different denominators by finding a common denominator and then adding or subtracting the fractions.

Q: How do I add or subtract fractions with different denominators?

A: To add or subtract fractions with different denominators, follow these steps:

  1. Find the LCM of the denominators.
  2. Rewrite each fraction with the LCM as the denominator.
  3. Add or subtract the fractions.

Q: Can I simplify a fraction after adding or subtracting it?

A: Yes, you can simplify a fraction after adding or subtracting it by dividing both the numerator and the denominator by their GCD.

Q: What are some real-world applications of combining fractions with different denominators?

A: Combining fractions with different denominators has numerous real-world applications in various fields, including science, engineering, and finance. For example, in physics, combining fractions is used to calculate the acceleration of an object. In engineering, combining fractions is used to calculate the stress on a material. In finance, combining fractions is used to calculate the interest rate on a loan.

Conclusion

In conclusion, combining fractions with different denominators is a fundamental concept in mathematics that requires a thorough understanding of algebraic manipulations. By finding a common denominator and then adding or subtracting the fractions, we can simplify complex fractions and solve various mathematical problems. We hope that this Q&A article has provided you with a better understanding of combining fractions with different denominators.

Common Denominators and Least Common Multiples: Practice Problems

  1. Find the LCM of 8 and 12.
  2. Find the GCD of 12 and 18.
  3. Add the fractions 14\frac{1}{4} and 16\frac{1}{6}.
  4. Subtract the fractions 34\frac{3}{4} and 23\frac{2}{3}.
  5. Simplify the fraction 1520\frac{15}{20}.

Answers

  1. The LCM of 8 and 12 is 24.
  2. The GCD of 12 and 18 is 6.
  3. The sum of the fractions is 512\frac{5}{12}.
  4. The difference of the fractions is 512\frac{5}{12}.
  5. The simplified fraction is 34\frac{3}{4}.