Express In Lowest Terms: 27 A 3 − 1 9 A 2 − 6 A + 1 \frac{27 A^3-1}{9 A^2-6 A+1} 9 A 2 − 6 A + 1 27 A 3 − 1 ​

Introduction

In algebra, expressing fractions in lowest terms is an essential skill that helps simplify complex expressions and solve equations. When dealing with algebraic fractions, it's crucial to factorize the numerator and denominator to identify any common factors that can be canceled out. In this article, we will explore the process of expressing the algebraic fraction $\frac{27 a^3-1}{9 a^2-6 a+1}$ in lowest terms.

Understanding the Concept of Lowest Terms

A fraction is said to be in lowest terms when the numerator and denominator have no common factors other than 1. In other words, the fraction cannot be simplified further by canceling out any common factors. To express a fraction in lowest terms, we need to factorize the numerator and denominator and identify any common factors that can be canceled out.

Step 1: Factorize the Numerator and Denominator

To factorize the numerator and denominator, we need to identify any common factors that can be factored out. In this case, the numerator $27 a^3-1$ can be factored as $(3 a-1)(9 a^2+3 a+1)$, and the denominator $9 a^2-6 a+1$ can be factored as $(3 a-1)^2$.

import sympy as sp
a = sp.symbols('a')

numerator = 27a**3 - 1
denominator = 9a**2 - 6*a + 1

numerator_factors = sp.factor(numerator)
denominator_factors = sp.factor(denominator)
print("Numerator factors:", numerator_factors)
print("Denominator factors:", denominator_factors)

Step 2: Identify Common Factors

Once we have factored the numerator and denominator, we need to identify any common factors that can be canceled out. In this case, we can see that both the numerator and denominator have a common factor of $(3 a-1)$.

Step 3: Cancel Out Common Factors

Now that we have identified the common factor, we can cancel it out to simplify the fraction. To cancel out the common factor, we need to divide both the numerator and denominator by the common factor.

# Cancel out the common factor
simplified_fraction = (numerator_factors / (3*a-1)) / (denominator_factors / (3*a-1))
print("Simplified fraction:", simplified_fraction)

Conclusion

Expressing algebraic fractions in lowest terms is an essential skill that helps simplify complex expressions and solve equations. By following the steps outlined in this article, we can express the algebraic fraction $\frac{27 a^3-1}{9 a^2-6 a+1}$ in lowest terms. The simplified fraction is $\frac{9 a^2+3 a+1}{(3 a-1)^2}$.

Final Answer

Introduction

In our previous article, we explored the process of expressing the algebraic fraction $\frac{27 a^3-1}{9 a^2-6 a+1}$ in lowest terms. In this article, we will answer some frequently asked questions related to expressing algebraic fractions in lowest terms.

Q: What is the purpose of expressing algebraic fractions in lowest terms?

A: Expressing algebraic fractions in lowest terms is an essential skill that helps simplify complex expressions and solve equations. By simplifying fractions, we can make it easier to perform calculations and arrive at the correct solution.

Q: How do I know if a fraction is in lowest terms?

A: A fraction is said to be in lowest terms when the numerator and denominator have no common factors other than 1. To check if a fraction is in lowest terms, you can factorize the numerator and denominator and see if there are any common factors that can be canceled out.

Q: What is the difference between simplifying a fraction and expressing it in lowest terms?

A: Simplifying a fraction involves reducing it to its simplest form by canceling out any common factors. Expressing a fraction in lowest terms involves identifying any common factors that can be canceled out and simplifying the fraction to its simplest form.

Q: Can I use a calculator to simplify fractions?

A: Yes, you can use a calculator to simplify fractions. However, it's essential to understand the underlying math concepts and be able to simplify fractions manually. This will help you to identify any errors or issues with the calculator's output.

Q: How do I factorize the numerator and denominator of a fraction?

A: To factorize the numerator and denominator of a fraction, you can use the following steps:

1. Identify any common factors that can be factored out.
2. Use the distributive property to factorize the numerator and denominator.
3. Look for any patterns or relationships between the factors.

Q: What are some common mistakes to avoid when simplifying fractions?

A: Some common mistakes to avoid when simplifying fractions include:

1. Not canceling out common factors.
2. Not simplifying the fraction to its simplest form.
3. Not checking for any errors or issues with the calculator's output.

Q: Can I use algebraic identities to simplify fractions?

A: Yes, you can use algebraic identities to simplify fractions. Algebraic identities are mathematical formulas that can be used to simplify expressions and equations. By using algebraic identities, you can simplify fractions and arrive at the correct solution.

Q: How do I know if a fraction is irreducible?

A: A fraction is said to be irreducible when it cannot be simplified further by canceling out any common factors. To check if a fraction is irreducible, you can factorize the numerator and denominator and see if there are any common factors that can be canceled out.

Conclusion

Expressing algebraic fractions in lowest terms is an essential skill that helps simplify complex expressions and solve equations. By following the steps outlined in this article and avoiding common mistakes, you can simplify fractions and arrive at the correct solution.

Final Answer

The final answer is $\boxed{\frac{9 a^2+3 a+1}{(3 a-1)^2}}$.