Perform The Indicated Operation And Simplify The Result.$\[ \frac{12 A^2 - 3}{2} \cdot (2a + 1)^{-2} \cdot \left(\frac{6}{2a + 1}\right)^{-1} = \\]

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will explore the process of simplifying a complex algebraic expression, step by step. We will use the given expression as an example and break it down into manageable parts, making it easier to understand and simplify.

The Given Expression

The given expression is:

12a2−32⋅(2a+1)−2⋅(62a+1)−1={ \frac{12 a^2 - 3}{2} \cdot (2a + 1)^{-2} \cdot \left(\frac{6}{2a + 1}\right)^{-1} = }

This expression involves several operations, including multiplication, division, and exponentiation. Our goal is to simplify this expression and arrive at a final result.

Step 1: Simplify the Numerator

The numerator of the expression is 12a2−312 a^2 - 3. We can simplify this expression by factoring out the common term 33:

12a2−3=3(4a2−1){ 12 a^2 - 3 = 3(4a^2 - 1) }

This simplification will help us in the next step.

Step 2: Simplify the Denominator

The denominator of the expression is 22. Since the denominator is a constant, we can simplify the expression by dividing the numerator by the denominator:

3(4a2−1)2=32(4a2−1){ \frac{3(4a^2 - 1)}{2} = \frac{3}{2}(4a^2 - 1) }

This simplification will help us in the next step.

Step 3: Simplify the Exponential Terms

The expression contains two exponential terms: (2a+1)−2(2a + 1)^{-2} and (62a+1)−1\left(\frac{6}{2a + 1}\right)^{-1}. We can simplify these terms by applying the rules of exponents:

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

(62a+1)−1=2a+16{ \left(\frac{6}{2a + 1}\right)^{-1} = \frac{2a + 1}{6} }

This simplification will help us in the next step.

Step 4: Multiply the Terms

Now that we have simplified the numerator, denominator, and exponential terms, we can multiply the terms together:

32(4a2−1)⋅1(2a+1)2⋅2a+16{ \frac{3}{2}(4a^2 - 1) \cdot \frac{1}{(2a + 1)^2} \cdot \frac{2a + 1}{6} }

We can cancel out the common terms (2a+1)(2a + 1) and 66:

32(4a2−1)⋅1(2a+1)2⋅2a+16=32(4a2−1)⋅16(2a+1){ \frac{3}{2}(4a^2 - 1) \cdot \frac{1}{(2a + 1)^2} \cdot \frac{2a + 1}{6} = \frac{3}{2}(4a^2 - 1) \cdot \frac{1}{6(2a + 1)} }

This simplification will help us in the next step.

Step 5: Simplify the Final Expression

We can simplify the final expression by canceling out the common terms:

32(4a2−1)⋅16(2a+1)=312(4a2−1)⋅12a+1{ \frac{3}{2}(4a^2 - 1) \cdot \frac{1}{6(2a + 1)} = \frac{3}{12}(4a^2 - 1) \cdot \frac{1}{2a + 1} }

We can simplify the fraction 312\frac{3}{12} by dividing both the numerator and denominator by 33:

312(4a2−1)⋅12a+1=14(4a2−1)⋅12a+1{ \frac{3}{12}(4a^2 - 1) \cdot \frac{1}{2a + 1} = \frac{1}{4}(4a^2 - 1) \cdot \frac{1}{2a + 1} }

This is the final simplified expression.

Conclusion

Simplifying algebraic expressions can be a challenging task, but by breaking it down into manageable parts and applying the rules of algebra, we can arrive at a final result. In this article, we used the given expression as an example and simplified it step by step. We hope that this article has provided a clear and concise guide to simplifying algebraic expressions.

Final Answer

The final simplified expression is:

14(4a2−1)⋅12a+1{ \frac{1}{4}(4a^2 - 1) \cdot \frac{1}{2a + 1} }

This expression cannot be simplified further.

Discussion

The given expression involves several operations, including multiplication, division, and exponentiation. We used the rules of algebra to simplify the expression step by step. The final simplified expression is a fraction with two terms in the numerator and one term in the denominator.

Key Takeaways

  • Simplifying algebraic expressions involves breaking down the expression into manageable parts and applying the rules of algebra.
  • The rules of exponents can be used to simplify exponential terms.
  • Multiplying and dividing fractions can be simplified by canceling out common terms.
  • The final simplified expression may involve fractions with multiple terms in the numerator and denominator.

Common Mistakes

  • Failing to simplify the numerator and denominator separately.
  • Failing to apply the rules of exponents correctly.
  • Failing to cancel out common terms when multiplying and dividing fractions.

Real-World Applications

Simplifying algebraic expressions has numerous real-world applications, including:

  • Calculating the area and perimeter of shapes.
  • Determining the volume of solids.
  • Modeling population growth and decay.
  • Solving systems of linear equations.

Conclusion

Introduction

In our previous article, we explored the process of simplifying algebraic expressions step by step. We used a complex expression as an example and broke it down into manageable parts, making it easier to understand and simplify. In this article, we will answer some frequently asked questions about simplifying algebraic expressions.

Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to identify the numerator and denominator. If the numerator and denominator are both polynomials, we can simplify the expression by factoring out common terms.

Q: How do I simplify exponential terms in an algebraic expression?

A: To simplify exponential terms in an algebraic expression, we can apply the rules of exponents. For example, if we have the term (2a+1)−2(2a + 1)^{-2}, we can simplify it by writing it as 1(2a+1)2\frac{1}{(2a + 1)^2}.

Q: Can I simplify an algebraic expression by canceling out common terms?

A: Yes, you can simplify an algebraic expression by canceling out common terms. For example, if we have the expression 32(4a2−1)⋅1(2a+1)2⋅2a+16\frac{3}{2}(4a^2 - 1) \cdot \frac{1}{(2a + 1)^2} \cdot \frac{2a + 1}{6}, we can cancel out the common terms (2a+1)(2a + 1) and 66.

Q: What is the difference between simplifying an algebraic expression and solving an equation?

A: Simplifying an algebraic expression involves reducing the expression to its simplest form, while solving an equation involves finding the value of the variable that makes the equation true. For example, if we have the equation 2x+3=52x + 3 = 5, we can simplify the expression 2x+32x + 3 to 2x+12x + 1, but we still need to solve for xx.

Q: Can I use a calculator to simplify an algebraic expression?

A: Yes, you can use a calculator to simplify an algebraic expression. However, it's always a good idea to check your work by simplifying the expression manually.

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

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

  • Failing to simplify the numerator and denominator separately.
  • Failing to apply the rules of exponents correctly.
  • Failing to cancel out common terms when multiplying and dividing fractions.

Q: How do I know if an algebraic expression is already simplified?

A: To determine if an algebraic expression is already simplified, you can try to simplify it further by applying the rules of algebra. If you cannot simplify the expression any further, then it is already simplified.

Q: Can I use algebraic expressions to solve real-world problems?

A: Yes, you can use algebraic expressions to solve real-world problems. For example, you can use algebraic expressions to calculate the area and perimeter of shapes, determine the volume of solids, model population growth and decay, and solve systems of linear equations.

Conclusion

Simplifying algebraic expressions is an essential skill for any math enthusiast. By understanding the rules of algebra and applying them correctly, you can simplify complex expressions and arrive at a final result. We hope that this article has provided a clear and concise guide to simplifying algebraic expressions and answering some frequently asked questions.

Additional Resources

If you're looking for additional resources to help you simplify algebraic expressions, here are a few suggestions:

  • Khan Academy: Algebra
  • Mathway: Algebra Calculator
  • Wolfram Alpha: Algebra Solver

Final Tips

  • Practice simplifying algebraic expressions regularly to build your skills and confidence.
  • Use online resources and calculators to check your work and get help when you need it.
  • Don't be afraid to ask for help if you're struggling with a particular expression or concept.

By following these tips and practicing regularly, you'll be well on your way to becoming a master of simplifying algebraic expressions!