Simplify The Expression: ${ \frac{7 \cdot 2^x - 3 \cdot 2 {x+1}}{2 X - 2^{x-1}} }$

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 involving exponents and fractions. We will break down the problem into manageable steps, and by the end of this guide, you will be able to simplify the expression with ease.

The Given Expression

The expression we need to simplify is:

$$\frac{7 \cdot 2^x - 3 \cdot 2^{x+1}}{2^x - 2^{x-1}}$$

Step 1: Factor Out Common Terms

To simplify the expression, we need to factor out common terms from the numerator and denominator. Let's start by factoring out $2^x$ from the numerator:

$$\frac{7 \cdot 2^x - 3 \cdot 2^{x+1}}{2^x - 2^{x-1}} = \frac{2^x(7 - 3 \cdot 2)}{2^x - 2^{x-1}}$$

Step 2: Simplify the Numerator

Now, let's simplify the numerator by evaluating the expression inside the parentheses:

$$\frac{2^x(7 - 3 \cdot 2)}{2^x - 2^{x-1}} = \frac{2^x(7 - 6)}{2^x - 2^{x-1}}$$

Step 3: Simplify the Expression Inside the Parentheses

The expression inside the parentheses can be simplified as follows:

$$\frac{2^x(7 - 6)}{2^x - 2^{x-1}} = \frac{2^x(1)}{2^x - 2^{x-1}}$$

Step 4: Factor Out Common Terms from the Denominator

Now, let's factor out common terms from the denominator:

$$\frac{2^x(1)}{2^x - 2^{x-1}} = \frac{2^x(1)}{2^x(1 - 2^{-1})}$$

Step 5: Simplify the Expression Inside the Parentheses

The expression inside the parentheses can be simplified as follows:

$$\frac{2^x(1)}{2^x(1 - 2^{-1})} = \frac{2^x(1)}{2^x(\frac{1}{2})}$$

Step 6: Cancel Out Common Terms

Now, let's cancel out common terms:

$$\frac{2^x(1)}{2^x(\frac{1}{2})} = \frac{1}{\frac{1}{2}}$$

Step 7: Simplify the Final Expression

The final expression can be simplified as follows:

$$\frac{1}{\frac{1}{2}} = 2$$

Conclusion

In this article, we have simplified the given expression involving exponents and fractions. We have broken down the problem into manageable steps, and by the end of this guide, you will be able to simplify the expression with ease. Remember to factor out common terms, simplify the numerator and denominator, and cancel out common terms to arrive at the final answer.

Frequently Asked Questions

• Q: What is the final answer to the given expression? A: The final answer is 2.
• Q: How do I simplify the expression? A: To simplify the expression, factor out common terms, simplify the numerator and denominator, and cancel out common terms.
• Q: What are the steps involved in simplifying the expression? A: The steps involved in simplifying the expression are:
  1. Factor out common terms
  2. Simplify the numerator
  3. Simplify the expression inside the parentheses
  4. Factor out common terms from the denominator
  5. Simplify the expression inside the parentheses
  6. Cancel out common terms
  7. Simplify the final expression

Additional Resources

• For more information on algebraic manipulation, visit the Khan Academy website.
• For more practice problems, visit the Mathway website.
• For more resources on mathematics, visit the Wolfram Alpha website.

Final Thoughts

Simplifying expressions is an essential skill that every student and professional should possess. By following the steps outlined in this article, you will be able to simplify the expression with ease. Remember to factor out common terms, simplify the numerator and denominator, and cancel out common terms to arrive at the final answer.

Introduction

In our previous article, we simplified the expression $\frac{7 \cdot 2^x - 3 \cdot 2^{x+1}}{2^x - 2^{x-1}}$ using algebraic manipulation. In this article, we will provide a Q&A guide to help you understand the steps involved in simplifying the expression.

Q&A Guide

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

A: The first step in simplifying the expression is to factor out common terms from the numerator and denominator.

Q: How do I factor out common terms?

A: To factor out common terms, look for the greatest common factor (GCF) of the terms in the numerator and denominator. In this case, the GCF is $2^x$.

Q: What is the next step in simplifying the expression?

A: The next step is to simplify the numerator by evaluating the expression inside the parentheses.

Q: How do I simplify the numerator?

A: To simplify the numerator, evaluate the expression inside the parentheses. In this case, the expression inside the parentheses is $7 - 3 \cdot 2$.

Q: What is the result of simplifying the numerator?

A: The result of simplifying the numerator is $7 - 6 = 1$.

Q: What is the next step in simplifying the expression?

A: The next step is to factor out common terms from the denominator.

Q: How do I factor out common terms from the denominator?

A: To factor out common terms from the denominator, look for the greatest common factor (GCF) of the terms in the denominator. In this case, the GCF is $2^x$.

Q: What is the result of factoring out common terms from the denominator?

A: The result of factoring out common terms from the denominator is $2^x(1 - 2^{-1})$.

Q: What is the next step in simplifying the expression?

A: The next step is to simplify the expression inside the parentheses.

Q: How do I simplify the expression inside the parentheses?

A: To simplify the expression inside the parentheses, evaluate the expression $1 - 2^{-1}$.

Q: What is the result of simplifying the expression inside the parentheses?

A: The result of simplifying the expression inside the parentheses is $\frac{1}{2}$.

Q: What is the next step in simplifying the expression?

A: The next step is to cancel out common terms.

Q: How do I cancel out common terms?

A: To cancel out common terms, look for terms that are common to both the numerator and denominator. In this case, the common term is $2^x$.

Q: What is the result of canceling out common terms?

A: The result of canceling out common terms is $\frac{1}{\frac{1}{2}}$.

Q: What is the final step in simplifying the expression?

A: The final step is to simplify the final expression.

Q: How do I simplify the final expression?

A: To simplify the final expression, evaluate the expression $\frac{1}{\frac{1}{2}}$.

Q: What is the result of simplifying the final expression?

A: The result of simplifying the final expression is $2$.

Conclusion

In this Q&A guide, we have provided step-by-step instructions on how to simplify the expression $\frac{7 \cdot 2^x - 3 \cdot 2^{x+1}}{2^x - 2^{x-1}}$. We have covered the steps involved in simplifying the expression, including factoring out common terms, simplifying the numerator and denominator, and canceling out common terms. By following these steps, you will be able to simplify the expression with ease.

Frequently Asked Questions

• Q: What is the final answer to the given expression? A: The final answer is 2.
• Q: How do I simplify the expression? A: To simplify the expression, factor out common terms, simplify the numerator and denominator, and cancel out common terms.
• Q: What are the steps involved in simplifying the expression? A: The steps involved in simplifying the expression are:
  1. Factor out common terms
  2. Simplify the numerator
  3. Simplify the expression inside the parentheses
  4. Factor out common terms from the denominator
  5. Simplify the expression inside the parentheses
  6. Cancel out common terms
  7. Simplify the final expression

Additional Resources

• For more information on algebraic manipulation, visit the Khan Academy website.
• For more practice problems, visit the Mathway website.
• For more resources on mathematics, visit the Wolfram Alpha website.

Final Thoughts

Simplifying expressions is an essential skill that every student and professional should possess. By following the steps outlined in this Q&A guide, you will be able to simplify the expression with ease. Remember to factor out common terms, simplify the numerator and denominator, and cancel out common terms to arrive at the final answer.