Simplify The Expression: $\[ \frac{a-\frac{2a-1}{a}}{\frac{1-a}{3a}} \\]

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Introduction

Algebraic manipulation is a crucial skill in mathematics, and simplifying expressions is an essential part of it. In this article, we will focus on simplifying a complex expression involving fractions and variables. We will break down the expression into manageable steps, using algebraic properties and techniques to simplify it.

The Expression to Simplify

The given expression is:

a−2a−1a1−a3a\frac{a-\frac{2a-1}{a}}{\frac{1-a}{3a}}

This expression involves fractions within fractions, making it challenging to simplify. Our goal is to simplify this expression to its simplest form.

Step 1: Simplify the Fraction within the Fraction

To simplify the expression, we start by simplifying the fraction within the fraction. We have:

2a−1a\frac{2a-1}{a}

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 1. However, we can rewrite the fraction as:

2a−1a=2aa−1a=2−1a\frac{2a-1}{a} = \frac{2a}{a} - \frac{1}{a} = 2 - \frac{1}{a}

Step 2: Substitute the Simplified Fraction

Now that we have simplified the fraction within the fraction, we can substitute it back into the original expression:

a−2a−1a1−a3a=a−(2−1a)1−a3a\frac{a-\frac{2a-1}{a}}{\frac{1-a}{3a}} = \frac{a-(2 - \frac{1}{a})}{\frac{1-a}{3a}}

Step 3: Simplify the Expression Further

We can simplify the expression further by combining like terms in the numerator:

a−(2−1a)1−a3a=a−2+1a1−a3a\frac{a-(2 - \frac{1}{a})}{\frac{1-a}{3a}} = \frac{a-2 + \frac{1}{a}}{\frac{1-a}{3a}}

Step 4: Simplify the Numerator

We can simplify the numerator by combining the terms:

a−2+1a1−a3a=a2−2a+1a1−a3a\frac{a-2 + \frac{1}{a}}{\frac{1-a}{3a}} = \frac{\frac{a^2-2a+1}{a}}{\frac{1-a}{3a}}

Step 5: Simplify the Expression Further

We can simplify the expression further by canceling out common factors in the numerator and the denominator:

a2−2a+1a1−a3a=a2−2a+1a⋅3a1−a\frac{\frac{a^2-2a+1}{a}}{\frac{1-a}{3a}} = \frac{a^2-2a+1}{a} \cdot \frac{3a}{1-a}

Step 6: Simplify the Expression Further

We can simplify the expression further by multiplying the numerators and the denominators:

a2−2a+1a⋅3a1−a=3a3−6a2+3aa(1−a)\frac{a^2-2a+1}{a} \cdot \frac{3a}{1-a} = \frac{3a^3-6a^2+3a}{a(1-a)}

Step 7: Simplify the Expression Further

We can simplify the expression further by canceling out common factors in the numerator and the denominator:

3a3−6a2+3aa(1−a)=3a(a2−2a+1)a(1−a)\frac{3a^3-6a^2+3a}{a(1-a)} = \frac{3a(a^2-2a+1)}{a(1-a)}

Step 8: Simplify the Expression Further

We can simplify the expression further by canceling out common factors in the numerator and the denominator:

3a(a2−2a+1)a(1−a)=3(a2−2a+1)1−a\frac{3a(a^2-2a+1)}{a(1-a)} = \frac{3(a^2-2a+1)}{1-a}

Step 9: Simplify the Expression Further

We can simplify the expression further by factoring the numerator:

3(a2−2a+1)1−a=3(a−1)21−a\frac{3(a^2-2a+1)}{1-a} = \frac{3(a-1)^2}{1-a}

Step 10: Simplify the Expression Further

We can simplify the expression further by canceling out common factors in the numerator and the denominator:

3(a−1)21−a=−3(a−1)2\frac{3(a-1)^2}{1-a} = -3(a-1)^2

Conclusion

In this article, we have simplified a complex expression involving fractions and variables. We have broken down the expression into manageable steps, using algebraic properties and techniques to simplify it. The final simplified expression is:

−3(a−1)2-3(a-1)^2

This expression is in its simplest form, and it cannot be simplified further.

Final Answer

The final answer is: −3(a−1)2\boxed{-3(a-1)^2}

Introduction

In our previous article, we simplified a complex expression involving fractions and variables. In this article, we will provide a Q&A guide to algebraic manipulation, focusing on the steps and techniques used to simplify the expression.

Q: What is algebraic manipulation?

A: Algebraic manipulation is the process of simplifying or transforming algebraic expressions using various techniques and properties.

Q: Why is algebraic manipulation important?

A: Algebraic manipulation is essential in mathematics, as it allows us to simplify complex expressions, solve equations, and understand mathematical concepts.

Q: What are some common techniques used in algebraic manipulation?

A: Some common techniques used in algebraic manipulation include:

  • Simplifying fractions
  • Canceling out common factors
  • Factoring expressions
  • Using algebraic properties (e.g., commutative, associative, distributive)

Q: How do I simplify a complex expression?

A: To simplify a complex expression, follow these steps:

  1. Identify the expression to simplify
  2. Break down the expression into manageable parts
  3. Use algebraic properties and techniques to simplify each part
  4. Combine the simplified parts to form the final expression

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

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

  • Not canceling out common factors
  • Not factoring expressions
  • Not using algebraic properties
  • Not checking for errors in the final expression

Q: How do I check for errors in the final expression?

A: To check for errors in the final expression, follow these steps:

  1. Review the original expression and the simplified expression
  2. Check for any errors in the algebraic properties or techniques used
  3. Verify that the final expression is in its simplest form

Q: What are some real-world applications of algebraic manipulation?

A: Algebraic manipulation has many real-world applications, including:

  • Physics and engineering: Algebraic manipulation is used to solve equations and simplify complex expressions in physics and engineering.
  • Computer science: Algebraic manipulation is used in computer science to simplify complex expressions and solve equations.
  • Economics: Algebraic manipulation is used in economics to simplify complex expressions and solve equations.

Q: Can I use algebraic manipulation to solve equations?

A: Yes, algebraic manipulation can be used to solve equations. By simplifying the equation and using algebraic properties and techniques, you can solve for the unknown variable.

Q: What are some common types of equations that can be solved using algebraic manipulation?

A: Some common types of equations that can be solved using algebraic manipulation include:

  • Linear equations
  • Quadratic equations
  • Polynomial equations

Conclusion

In this article, we have provided a Q&A guide to algebraic manipulation, focusing on the steps and techniques used to simplify complex expressions. We have also discussed common mistakes to avoid and real-world applications of algebraic manipulation.

Final Answer

The final answer is: Algebraic manipulation is a powerful tool for simplifying complex expressions and solving equations.

Additional Resources

  • For more information on algebraic manipulation, visit the following websites:
    • Khan Academy: Algebraic Manipulation
    • Mathway: Algebraic Manipulation
    • Wolfram Alpha: Algebraic Manipulation
  • For practice problems and exercises, visit the following websites:
    • IXL: Algebraic Manipulation
    • Math Open Reference: Algebraic Manipulation
    • Algebra.com: Algebraic Manipulation