Simplify:a. \[$\frac{15}{81}\$\]

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Introduction

Simplifying fractions is an essential skill in mathematics, and it's often used in various mathematical operations, such as adding, subtracting, multiplying, and dividing fractions. In this article, we will focus on simplifying the fraction $\frac{15}{81}$, which is a complex fraction that can be simplified using various techniques. We will explore the different methods of simplifying this fraction and provide step-by-step explanations to help you understand the process.

What is Simplifying a Fraction?

Simplifying a fraction involves reducing it to its lowest terms, which means expressing it as a ratio of the simplest form possible. This is done by finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both numbers by the GCD. The resulting fraction is the simplified form of the original fraction.

Method 1: Finding the Greatest Common Divisor (GCD)

To simplify the fraction $\frac{15}{81}$, we need to find the GCD of 15 and 81. The GCD is the largest number that divides both numbers without leaving a remainder. We can use various methods to find the GCD, such as listing the factors of each number or using the Euclidean algorithm.

Factors of 15

The factors of 15 are: 1, 3, 5, and 15.

Factors of 81

The factors of 81 are: 1, 3, 9, 27, and 81.

Finding the GCD

By comparing the factors of 15 and 81, we can see that the greatest common factor is 3. Therefore, the GCD of 15 and 81 is 3.

Simplifying the Fraction

Now that we have found the GCD, we can simplify the fraction $\frac{15}{81}$ by dividing both the numerator and the denominator by the GCD.

$\frac{15}{81} = \frac{15 \div 3}{81 \div 3} = \frac{5}{27}$

Therefore, the simplified form of the fraction $\frac{15}{81}$ is $\frac{5}{27}$.

Method 2: Using Prime Factorization

Another method of simplifying a fraction is by using prime factorization. This involves expressing both the numerator and the denominator as a product of prime numbers.

Prime Factorization of 15

The prime factorization of 15 is: $3 \times 5$

Prime Factorization of 81

The prime factorization of 81 is: $3^4$

Simplifying the Fraction

Now that we have expressed both the numerator and the denominator as a product of prime numbers, we can simplify the fraction by canceling out any common factors.

$\frac{15}{81} = \frac{3 \times 5}{3^4} = \frac{5}{3^3} = \frac{5}{27}$

Therefore, the simplified form of the fraction $\frac{15}{81}$ is $\frac{5}{27}$.

Conclusion

Simplifying fractions is an essential skill in mathematics, and it's often used in various mathematical operations. In this article, we have explored two methods of simplifying the fraction $\frac{15}{81}$: finding the greatest common divisor (GCD) and using prime factorization. We have provided step-by-step explanations to help you understand the process and have shown that the simplified form of the fraction $\frac{15}{81}$ is $\frac{5}{27}$.

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Introduction

In our previous article, we explored the process of simplifying the fraction $\frac{15}{81}$ using two different methods: finding the greatest common divisor (GCD) and using prime factorization. In this article, we will answer some frequently asked questions (FAQs) related to simplifying fractions, including the fraction $\frac{15}{81}$.

Q&A

Q: What is the greatest common divisor (GCD) of 15 and 81?

A: The greatest common divisor (GCD) of 15 and 81 is 3.

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

A: There are several methods to find the GCD of two numbers, including listing the factors of each number, using the Euclidean algorithm, or using a calculator.

Q: What is prime factorization?

A: Prime factorization is the process of expressing a number as a product of prime numbers.

Q: How do I simplify a fraction using prime factorization?

A: To simplify a fraction using prime factorization, you need to express both the numerator and the denominator as a product of prime numbers and then cancel out any common factors.

Q: Why is it important to simplify fractions?

A: Simplifying fractions is important because it helps to reduce the complexity of mathematical operations and makes it easier to compare and manipulate fractions.

Q: Can I simplify a fraction that has a variable in the numerator or denominator?

A: Yes, you can simplify a fraction that has a variable in the numerator or denominator by following the same steps as simplifying a fraction with numerical values.

Q: How do I know if a fraction is already in its simplest form?

A: A fraction is already in its simplest form if the numerator and denominator have no common factors other than 1.

Q: Can I use a calculator to simplify fractions?

A: Yes, you can use a calculator to simplify fractions, but it's also important to understand the underlying mathematical concepts and be able to simplify fractions manually.

Q: What is the simplified form of the fraction $\frac{15}{81}$?

A: The simplified form of the fraction $\frac{15}{81}$ is $\frac{5}{27}$.

Additional Tips and Resources

• To simplify fractions, it's essential to understand the concept of greatest common divisor (GCD) and prime factorization.
• You can use online resources, such as calculators or math websites, to help you simplify fractions.
• Practice simplifying fractions with different numerical values and variables to become more comfortable with the process.
• If you're struggling to simplify a fraction, try breaking it down into smaller steps or using a different method, such as prime factorization.

Conclusion

Simplifying fractions is an essential skill in mathematics, and it's often used in various mathematical operations. In this article, we have answered some frequently asked questions (FAQs) related to simplifying fractions, including the fraction $\frac{15}{81}$. We hope that this article has provided you with a better understanding of the process of simplifying fractions and has helped you to become more confident in your ability to simplify fractions.