Simplify The Expression: ${ \frac{5 \cdot 2^x - 4 \cdot 2 {x-2}}{2 X - 2^{x-1}} }$

Mar 9, 2025 by ADMIN 84 views

Simplify the Expression: A Step-by-Step Guide to Algebraic Manipulation

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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 given expression involving exponents and fractions. We will break down the problem step by step, using algebraic properties and techniques to arrive at a simplified expression.

The Given Expression

The given expression is:

\frac{5 \cdot 2^x - 4 \cdot 2^{x-2}}{2^x - 2^{x-1}}

This expression involves exponents, fractions, and algebraic operations. Our goal is to simplify this expression by manipulating the terms and using algebraic properties.

Step 1: Factor Out Common Terms

The first step in simplifying the expression is to factor out common terms. We can start by factoring out the common term $2^x$ from the numerator:

\frac{5 \cdot 2^x - 4 \cdot 2^{x-2}}{2^x - 2^{x-1}} = \frac{2^x(5 - 4 \cdot 2^{-2})}{2^x - 2^{x-1}}

Step 2: Simplify the Numerator

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

5 - 4 \cdot 2^{-2} = 5 - \frac{4}{2^2} = 5 - \frac{4}{4} = 5 - 1 = 4

So, the expression becomes:

\frac{2^x(4)}{2^x - 2^{x-1}}

Step 3: Simplify the Denominator

Next, let's simplify the denominator by factoring out the common term $2^{x-1}$ :

2^x - 2^{x-1} = 2^{x-1}(2 - 1) = 2^{x-1}

So, the expression becomes:

\frac{2^x(4)}{2^{x-1}}

Step 4: Cancel Out Common Factors

Now, we can cancel out the common factor $2^{x-1}$ from the numerator and denominator:

\frac{2^x(4)}{2^{x-1}} = 2^{x-(x-1)} \cdot 4 = 2^1 \cdot 4 = 2 \cdot 4 = 8

Conclusion

In this article, we simplified the given expression by factoring out common terms, simplifying the numerator and denominator, and canceling out common factors. The final simplified expression is:

8

This expression is much simpler than the original expression, and it can be used as a building block for further algebraic manipulations.

Tips and Tricks

When simplifying expressions, always look for common factors to factor out.
Use algebraic properties, such as the distributive property and the commutative property, to simplify expressions.
Cancel out common factors to simplify expressions.
Use exponent rules, such as the product rule and the power rule, to simplify expressions involving exponents.

Practice Problems

Simplify the expression: $\frac{3 \cdot 2^x - 2 \cdot 2^{x-1}}{2^x - 2^{x-1}}$
Simplify the expression: $\frac{4 \cdot 2^x - 3 \cdot 2^{x-2}}{2^x - 2^{x-1}}$
Simplify the expression: $\frac{2 \cdot 2^x - 3 \cdot 2^{x-1}}{2^x - 2^{x-1}}$

These practice problems will help you reinforce your understanding of algebraic manipulation and simplifying expressions.

Final Thoughts

Simplifying expressions is an essential skill in mathematics, and it requires practice and patience. By following the steps outlined in this article, you can simplify complex expressions and arrive at a simplified expression. Remember to always look for common factors, use algebraic properties, and cancel out common factors to simplify expressions. With practice, you will become proficient in simplifying expressions and solving algebraic problems.

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Introduction

In our previous article, we simplified a given expression involving exponents and fractions. We broke down the problem step by step, using algebraic properties and techniques to arrive at a simplified expression. In this article, we will answer some frequently asked questions (FAQs) related to simplifying expressions and algebraic manipulation.

Q&A

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

A: The first step in simplifying an 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 an expression with exponents?

A: To simplify an expression with exponents, use the exponent rules, such as the product rule and the power rule. For example, if you have an expression like $2^x \cdot 2^y$ , you can simplify it by adding the exponents: $2^{x+y}$ .

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

A: Simplifying an expression involves reducing the complexity of the expression by combining like terms, factoring out common factors, and canceling out common factors. Solving an equation, on the other hand, involves finding the value of the variable that makes the equation true.

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

A: You should simplify an expression when it is necessary to make the expression more manageable or to make it easier to solve. For example, if you have an expression like $\frac{2x + 3}{x + 1}$ , you can simplify it by factoring out the common factor $x + 1$ : $\frac{2(x + 1) + 1}{x + 1}$ .

Q: Can I simplify an expression with variables in the exponent?

A: Yes, you can simplify an expression with variables in the exponent. For example, if you have an expression like $2^{x+1}$ , you can simplify it by using the exponent rule: $2^{x+1} = 2^x \cdot 2^1$ .

Q: How do I simplify an expression with fractions?

A: To simplify an expression with fractions, you can multiply the numerator and denominator by the same value to eliminate the fraction. For example, if you have an expression like $\frac{2}{3}$ , you can simplify it by multiplying the numerator and denominator by 3: $\frac{2 \cdot 3}{3 \cdot 3} = \frac{6}{9}$ .

Tips and Tricks

Always look for common factors to factor out.
Use algebraic properties, such as the distributive property and the commutative property, to simplify expressions.
Cancel out common factors to simplify expressions.
Use exponent rules, such as the product rule and the power rule, to simplify expressions involving exponents.
Simplify expressions when it is necessary to make the expression more manageable or to make it easier to solve.

Practice Problems

Simplify the expression: $\frac{3x + 2}{x + 1}$
Simplify the expression: $\frac{2x^2 + 3x}{x + 1}$
Simplify the expression: $\frac{x^2 + 2x}{x + 1}$

These practice problems will help you reinforce your understanding of algebraic manipulation and simplifying expressions.

Final Thoughts

Simplifying expressions is an essential skill in mathematics, and it requires practice and patience. By following the steps outlined in this article and using the tips and tricks provided, you can simplify complex expressions and arrive at a simplified expression. Remember to always look for common factors, use algebraic properties, and cancel out common factors to simplify expressions. With practice, you will become proficient in simplifying expressions and solving algebraic problems.