Simplify The Expression:${ \frac{2^{n+4} - 6 \times 2 {n+1}}{2 {2(n+1)}} }$

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Introduction

Algebraic manipulation is a crucial aspect of mathematics, and simplifying expressions is an essential skill that every student and professional should possess. In this article, we will focus on simplifying a given expression, which involves exponentiation and algebraic operations. We will break down the expression into manageable steps, and by the end of this article, you will have a clear understanding of how to simplify complex expressions.

The Given Expression

The given expression is:

2n+4−6×2n+122(n+1)\frac{2^{n+4} - 6 \times 2^{n+1}}{2^{2(n+1)}}

This expression involves exponentiation, multiplication, and division. Our goal is to simplify this expression to its most basic form.

Step 1: Simplify the Exponents

To simplify the expression, we start by simplifying the exponents. We can rewrite the expression as:

2n+4−6×2n+122(n+1)=2n×24−6×2n×2122n+2\frac{2^{n+4} - 6 \times 2^{n+1}}{2^{2(n+1)}} = \frac{2^{n} \times 2^4 - 6 \times 2^{n} \times 2^1}{2^{2n+2}}

Step 2: Apply the Laws of Exponents

We can now apply the laws of exponents to simplify the expression further. We know that am×an=am+na^m \times a^n = a^{m+n} and aman=am−n\frac{a^m}{a^n} = a^{m-n}. Applying these laws, we get:

2n×24−6×2n×2122n+2=2n+4−6×2n+122n+2\frac{2^{n} \times 2^4 - 6 \times 2^{n} \times 2^1}{2^{2n+2}} = \frac{2^{n+4} - 6 \times 2^{n+1}}{2^{2n+2}}

Step 3: Factor Out Common Terms

We can now factor out common terms from the numerator. We can rewrite the expression as:

2n+4−6×2n+122n+2=2n+1(23−6)22n+2\frac{2^{n+4} - 6 \times 2^{n+1}}{2^{2n+2}} = \frac{2^{n+1}(2^3 - 6)}{2^{2n+2}}

Step 4: Simplify the Numerator

We can now simplify the numerator by evaluating the expression inside the parentheses. We get:

23−6=8−6=22^3 - 6 = 8 - 6 = 2

So, the expression becomes:

2n+1(23−6)22n+2=2n+1×222n+2\frac{2^{n+1}(2^3 - 6)}{2^{2n+2}} = \frac{2^{n+1} \times 2}{2^{2n+2}}

Step 5: Apply the Laws of Exponents Again

We can now apply the laws of exponents again to simplify the expression further. We get:

2n+1×222n+2=2n+1+122n+2=2n+222n+2\frac{2^{n+1} \times 2}{2^{2n+2}} = \frac{2^{n+1+1}}{2^{2n+2}} = \frac{2^{n+2}}{2^{2n+2}}

Step 6: Simplify the Expression

We can now simplify the expression by applying the laws of exponents. We get:

2n+222n+2=2(n+2)−(2n+2)=2−n\frac{2^{n+2}}{2^{2n+2}} = 2^{(n+2)-(2n+2)} = 2^{-n}

And that's it! We have successfully simplified the given expression.

Conclusion

Simplifying expressions is an essential skill in mathematics, and it requires a deep understanding of algebraic manipulation. In this article, we broke down the given expression into manageable steps and simplified it using the laws of exponents. We hope that this article has provided you with a clear understanding of how to simplify complex expressions.

Frequently Asked Questions

  • Q: What is the final simplified expression? A: The final simplified expression is 2−n2^{-n}.
  • Q: What are the laws of exponents? A: The laws of exponents are am×an=am+na^m \times a^n = a^{m+n} and aman=am−n\frac{a^m}{a^n} = a^{m-n}.
  • Q: How do I simplify expressions? A: To simplify expressions, you need to apply the laws of exponents, factor out common terms, and simplify the numerator and denominator.

Further Reading

If you want to learn more about algebraic manipulation and simplifying expressions, we recommend checking out the following resources:

  • Khan Academy: Algebraic Manipulation
  • Mathway: Simplifying Expressions
  • Wolfram Alpha: Algebraic Manipulation

We hope that this article has provided you with a clear understanding of how to simplify complex expressions. If you have any further questions or need help with algebraic manipulation, please don't hesitate to ask.

Introduction

In our previous article, we simplified the expression 2n+4−6×2n+122(n+1)\frac{2^{n+4} - 6 \times 2^{n+1}}{2^{2(n+1)}} using algebraic manipulation. In this article, we will answer some frequently asked questions related to simplifying expressions and provide additional resources for further learning.

Q&A

Q: What is the final simplified expression?

A: The final simplified expression is 2−n2^{-n}.

Q: What are the laws of exponents?

A: The laws of exponents are am×an=am+na^m \times a^n = a^{m+n} and aman=am−n\frac{a^m}{a^n} = a^{m-n}.

Q: How do I simplify expressions?

A: To simplify expressions, you need to apply the laws of exponents, factor out common terms, and simplify the numerator and denominator.

Q: What is the difference between simplifying expressions and solving equations?

A: Simplifying expressions involves reducing an expression to its most basic form, while solving equations involves finding the value of a variable that makes the equation true.

Q: Can I use a calculator to simplify expressions?

A: Yes, you can use a calculator to simplify expressions, but it's always a good idea to understand the underlying algebraic manipulation to ensure accuracy.

Q: How do I know when to simplify an expression?

A: You should simplify an expression when it's necessary to make the expression more manageable or to make it easier to solve an equation.

Q: Can I simplify expressions with variables in the exponent?

A: Yes, you can simplify expressions with variables in the exponent by applying the laws of exponents and simplifying the expression.

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

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

  • Not applying the laws of exponents correctly
  • Not factoring out common terms
  • Not simplifying the numerator and denominator
  • Not checking for errors in the final simplified expression

Additional Resources

If you want to learn more about algebraic manipulation and simplifying expressions, we recommend checking out the following resources:

  • Khan Academy: Algebraic Manipulation
  • Mathway: Simplifying Expressions
  • Wolfram Alpha: Algebraic Manipulation
  • MIT OpenCourseWare: Algebra
  • Algebra.com: Simplifying Expressions

Conclusion

Simplifying expressions is an essential skill in mathematics, and it requires a deep understanding of algebraic manipulation. In this article, we answered some frequently asked questions related to simplifying expressions and provided additional resources for further learning. We hope that this article has provided you with a clear understanding of how to simplify complex expressions.

Frequently Asked Questions (FAQs)

  • Q: What is the difference between simplifying expressions and solving equations? A: Simplifying expressions involves reducing an expression to its most basic form, while solving equations involves finding the value of a variable that makes the equation true.
  • Q: Can I use a calculator to simplify expressions? A: Yes, you can use a calculator to simplify expressions, but it's always a good idea to understand the underlying algebraic manipulation to ensure accuracy.
  • Q: How do I know when to simplify an expression? A: You should simplify an expression when it's necessary to make the expression more manageable or to make it easier to solve an equation.

Further Reading

If you want to learn more about algebraic manipulation and simplifying expressions, we recommend checking out the following resources:

  • Khan Academy: Algebraic Manipulation
  • Mathway: Simplifying Expressions
  • Wolfram Alpha: Algebraic Manipulation
  • MIT OpenCourseWare: Algebra
  • Algebra.com: Simplifying Expressions

We hope that this article has provided you with a clear understanding of how to simplify complex expressions. If you have any further questions or need help with algebraic manipulation, please don't hesitate to ask.