Simplify The Expression: ${ \frac{3 \cdot 2^m - 4 \cdot 2 {m-2}}{2 M - 2^{m-1}} }$

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Introduction

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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 given expression using algebraic techniques. The expression we will be working with is $\frac{3 \cdot 2^m - 4 \cdot 2^{m-2}}{2^m - 2^{m-1}}$. Our goal is to simplify this expression to its simplest form.

Understanding the Expression

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Before we start simplifying the expression, let's break it down and understand what it represents. The expression consists of two terms in the numerator and two terms in the denominator. The numerator is $3 \cdot 2^m - 4 \cdot 2^{m-2}$, and the denominator is $2^m - 2^{m-1}$.

Factoring Out Common Terms

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One of the first steps in simplifying the expression is to factor out common terms. In the numerator, we can factor out $2^{m-2}$ from both terms. This gives us $2^{m-2}(3 \cdot 2^2 - 4)$.

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

expr = (3 * 2m - 4 * 2(m-2)) / (2m - 2(m-1))

factored_expr = sp.factor(expr)
print(factored_expr)

Simplifying the Denominator

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Now that we have factored out common terms in the numerator, let's simplify the denominator. We can rewrite the denominator as $2^m - 2^{m-1} = 2^{m-1}(2 - 1) = 2^{m-1}$.

Combining the Simplified Numerator and Denominator

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Now that we have simplified the numerator and denominator, let's combine them. We can rewrite the expression as $\frac{2^{m-2}(3 \cdot 2^2 - 4)}{2^{m-1}}$.

Canceling Out Common Factors

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We can further simplify the expression by canceling out common factors. We can cancel out $2^{m-2}$ from the numerator and denominator, leaving us with $\frac{3 \cdot 2^2 - 4}{2}$.

Evaluating the Expression

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Now that we have simplified the expression, let's evaluate it. We can rewrite the expression as $\frac{12 - 4}{2} = \frac{8}{2} = 4$.

Conclusion

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In this article, we simplified the expression $\frac{3 \cdot 2^m - 4 \cdot 2^{m-2}}{2^m - 2^{m-1}}$ using algebraic techniques. We factored out common terms, simplified the denominator, combined the simplified numerator and denominator, canceled out common factors, and evaluated the expression. The final simplified expression is $4$.

Final Answer

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The final answer is $\boxed{4}$.

Step-by-Step Solution

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Here is the step-by-step solution to the problem:

1. Factor out common terms in the numerator: $2^{m-2}(3 \cdot 2^2 - 4)$
2. Simplify the denominator: $2^{m-1}$
3. Combine the simplified numerator and denominator: $\frac{2^{m-2}(3 \cdot 2^2 - 4)}{2^{m-1}}$
4. Cancel out common factors: $\frac{3 \cdot 2^2 - 4}{2}$
5. Evaluate the expression: $\frac{12 - 4}{2} = \frac{8}{2} = 4$

Frequently Asked Questions

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Q: What is the final simplified expression?

A: The final simplified expression is $4$.

Q: How do I simplify the expression?

A: To simplify the expression, you can factor out common terms, simplify the denominator, combine the simplified numerator and denominator, cancel out common factors, and evaluate the expression.

Q: What is the step-by-step solution to the problem?

A: The step-by-step solution to the problem is:

1. Factor out common terms in the numerator: $2^{m-2}(3 \cdot 2^2 - 4)$
2. Simplify the denominator: $2^{m-1}$
3. Combine the simplified numerator and denominator: $\frac{2^{m-2}(3 \cdot 2^2 - 4)}{2^{m-1}}$
4. Cancel out common factors: $\frac{3 \cdot 2^2 - 4}{2}$
5. Evaluate the expression: $\frac{12 - 4}{2} = \frac{8}{2} = 4$
   

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Introduction

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In our previous article, we simplified the expression $\frac{3 \cdot 2^m - 4 \cdot 2^{m-2}}{2^m - 2^{m-1}}$ using algebraic techniques. In this article, we will answer some frequently asked questions related to simplifying algebraic expressions.

Q&A

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

A: The first step in simplifying an algebraic expression is to factor out common terms. This involves identifying common factors in the numerator and denominator and factoring them out.

Q: How do I simplify a fraction with a variable in the denominator?

A: To simplify a fraction with a variable in the denominator, you can try to factor out the variable from the numerator and denominator. If the variable is not present in both the numerator and denominator, you can simplify the fraction by canceling out common factors.

Q: What is the difference between simplifying an expression and evaluating an expression?

A: Simplifying an expression involves rewriting it in a simpler form, while evaluating an expression involves finding its numerical value. For example, the expression $x^2 + 2x + 1$ can be simplified to $(x + 1)^2$, but its numerical value depends on the value of $x$.

Q: Can I simplify an expression with multiple variables?

A: Yes, you can simplify an expression with multiple variables. However, you need to be careful when factoring out common terms and canceling out common factors. It's also a good idea to use algebraic techniques such as substitution and elimination to simplify the expression.

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

A: An expression is already simplified if it cannot be rewritten in a simpler form using algebraic techniques. You can check if an expression is already simplified by trying to factor out common terms or cancel out common factors. If you cannot simplify the expression further, it is already simplified.

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

A: Yes, you can use a calculator to simplify an expression. However, it's always a good idea to check your work by hand to ensure that the calculator is giving you the correct answer.

Q: What are some common algebraic techniques used to simplify expressions?

A: Some common algebraic techniques used to simplify expressions include:

• Factoring out common terms
• Canceling out common factors
• Substitution
• Elimination
• Distributive property
• Combining like terms

Q: How do I know which algebraic technique to use?

A: The choice of algebraic technique depends on the specific expression you are working with. You can try different techniques and see which one works best for the expression.

Conclusion

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Simplifying algebraic expressions is an essential skill in mathematics. By understanding the different algebraic techniques and how to apply them, you can simplify complex expressions and make them easier to work with. Remember to always check your work by hand to ensure that the calculator is giving you the correct answer.

Final Answer

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The final answer is $\boxed{4}$.

Step-by-Step Solution

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Here is the step-by-step solution to the problem:

1. Factor out common terms in the numerator: $2^{m-2}(3 \cdot 2^2 - 4)$
2. Simplify the denominator: $2^{m-1}$
3. Combine the simplified numerator and denominator: $\frac{2^{m-2}(3 \cdot 2^2 - 4)}{2^{m-1}}$
4. Cancel out common factors: $\frac{3 \cdot 2^2 - 4}{2}$
5. Evaluate the expression: $\frac{12 - 4}{2} = \frac{8}{2} = 4$

Frequently Asked Questions

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Q: What is the final simplified expression?

A: The final simplified expression is $4$.

Q: How do I simplify the expression?

A: To simplify the expression, you can factor out common terms, simplify the denominator, combine the simplified numerator and denominator, cancel out common factors, and evaluate the expression.

Q: What is the step-by-step solution to the problem?

A: The step-by-step solution to the problem is:

1. Factor out common terms in the numerator: $2^{m-2}(3 \cdot 2^2 - 4)$
2. Simplify the denominator: $2^{m-1}$
3. Combine the simplified numerator and denominator: $\frac{2^{m-2}(3 \cdot 2^2 - 4)}{2^{m-1}}$
4. Cancel out common factors: $\frac{3 \cdot 2^2 - 4}{2}$
5. Evaluate the expression: $\frac{12 - 4}{2} = \frac{8}{2} = 4$