Simplify The Expression:${ \frac{5 {-x+1}-5 {-x-1}}{5 {-x+2}-5 {-x}} }$

Introduction

In algebra, simplifying expressions is a crucial skill that helps in solving complex problems and understanding mathematical concepts. The given expression, $\frac{5^{-x+1}-5^{-x-1}}{5^{-x+2}-5^{-x}}$, appears to be a challenging one, but with the right approach, it can be simplified to a more manageable form. In this article, we will delve into the world of algebraic manipulation and explore the steps required to simplify the given expression.

Understanding the Expression

Before we begin simplifying the expression, let's take a closer look at its components. The expression consists of two fractions in the numerator and denominator, each with a negative exponent. The numerator is $5^{-x+1}-5^{-x-1}$, and the denominator is $5^{-x+2}-5^{-x}$. To simplify this expression, we need to apply the rules of exponents and algebraic manipulation.

Simplifying the Numerator

Let's start by simplifying the numerator. We can rewrite the expression as $5^{-x+1}-5^{-x-1} = 5^{-x}(5^1-5^{-1})$. This simplification involves factoring out the common term $5^{-x}$ from both terms in the numerator.

Simplifying the Denominator

Next, let's simplify the denominator. We can rewrite the expression as $5^{-x+2}-5^{-x} = 5^{-x}(5^2-1)$. This simplification involves factoring out the common term $5^{-x}$ from both terms in the denominator.

Combining the Simplified Numerator and Denominator

Now that we have simplified the numerator and denominator, we can combine them to get the simplified expression. We have $\frac{5^{-x}(5^1-5^{-1})}{5^{-x}(5^2-1)}$. Notice that the common term $5^{-x}$ appears in both the numerator and denominator, so we can cancel it out.

Canceling Out the Common Term

After canceling out the common term $5^{-x}$, we are left with $\frac{5^1-5^{-1}}{5^2-1}$. This is the simplified expression.

Further Simplification

We can further simplify the expression by evaluating the exponents. We have $\frac{5-5^{-1}}{25-1}$. To evaluate the exponents, we need to remember that $5^{-1} = \frac{1}{5}$.

Evaluating the Exponents

Now that we have evaluated the exponents, we can simplify the expression further. We have $\frac{5-\frac{1}{5}}{24}$. To simplify this expression, we need to find a common denominator for the terms in the numerator.

Finding a Common Denominator

The common denominator for the terms in the numerator is 5. So, we can rewrite the expression as $\frac{\frac{25}{5}-\frac{1}{5}}{24}$. This simplification involves finding a common denominator for the terms in the numerator.

Simplifying the Numerator

Now that we have found a common denominator, we can simplify the numerator. We have $\frac{24}{5}$. This simplification involves combining the terms in the numerator.

Final Simplification

We are now left with $\frac{24}{5}$. This is the final simplified expression.

Conclusion

In this article, we have simplified the given expression using algebraic manipulation. We started by simplifying the numerator and denominator, and then combined them to get the simplified expression. We canceled out the common term $5^{-x}$ and evaluated the exponents to get the final simplified expression. This article demonstrates the importance of simplifying expressions in algebra and provides a step-by-step guide to simplifying complex expressions.

Frequently Asked Questions

• What is the simplified expression? The simplified expression is $\frac{24}{5}$.
• How do I simplify the expression? To simplify the expression, you need to apply the rules of exponents and algebraic manipulation. You can start by simplifying the numerator and denominator, and then combine them to get the simplified expression.
• What is the importance of simplifying expressions in algebra? Simplifying expressions in algebra is crucial in solving complex problems and understanding mathematical concepts. It helps in identifying patterns and relationships between variables, and makes it easier to solve equations and inequalities.

Final Thoughts

Simplifying expressions is a crucial skill in algebra that helps in solving complex problems and understanding mathematical concepts. By applying the rules of exponents and algebraic manipulation, we can simplify complex expressions and make them more manageable. This article provides a step-by-step guide to simplifying the given expression, and demonstrates the importance of simplifying expressions in algebra.

Introduction

In our previous article, we simplified the expression $\frac{5^{-x+1}-5^{-x-1}}{5^{-x+2}-5^{-x}}$ using algebraic manipulation. In this article, we will answer some frequently asked questions related to simplifying expressions in algebra.

Q&A

Q: What is the simplified expression?

A: The simplified expression is $\frac{24}{5}$.

Q: How do I simplify the expression?

A: To simplify the expression, you need to apply the rules of exponents and algebraic manipulation. You can start by simplifying the numerator and denominator, and then combine them to get the simplified expression.

Q: What is the importance of simplifying expressions in algebra?

A: Simplifying expressions in algebra is crucial in solving complex problems and understanding mathematical concepts. It helps in identifying patterns and relationships between variables, and makes it easier to solve equations and inequalities.

Q: How do I apply the rules of exponents?

A: To apply the rules of exponents, you need to remember the following rules:

• When multiplying two numbers with the same base, add their exponents.
• When dividing two numbers with the same base, subtract their exponents.
• When raising a number to a power, multiply the exponents.

Q: How do I simplify expressions with negative exponents?

A: To simplify expressions with negative exponents, you need to remember that a negative exponent is equal to the reciprocal of the base raised to the positive exponent. For example, $5^{-x} = \frac{1}{5^x}$.

Q: How do I simplify expressions with fractions?

A: To simplify expressions with fractions, you need to find a common denominator for the fractions and then combine them.

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

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

• Not applying the rules of exponents correctly
• Not simplifying the numerator and denominator separately
• Not canceling out common factors
• Not evaluating the exponents correctly

Q: How do I check my work when simplifying expressions?

A: To check your work when simplifying expressions, you need to:

• Simplify the expression step by step
• Check your work at each step
• Use a calculator to check your answer
• Compare your answer with the original expression

Conclusion

Simplifying expressions is a crucial skill in algebra that helps in solving complex problems and understanding mathematical concepts. By applying the rules of exponents and algebraic manipulation, we can simplify complex expressions and make them more manageable. This article provides a comprehensive guide to simplifying expressions in algebra, and answers some frequently asked questions related to this topic.

Final Thoughts

Simplifying expressions is a skill that takes practice to develop. With patience and persistence, you can become proficient in simplifying complex expressions and solving algebraic problems. Remember to apply the rules of exponents and algebraic manipulation correctly, and to check your work at each step.

Additional Resources

• Algebraic Manipulation: A Comprehensive Guide
• Simplifying Expressions: A Step-by-Step Guide
• Algebraic Identities: A List of Common Identities

Frequently Asked Questions

• What is the simplified expression? The simplified expression is $\frac{24}{5}$.
• How do I simplify the expression? To simplify the expression, you need to apply the rules of exponents and algebraic manipulation.
• What is the importance of simplifying expressions in algebra? Simplifying expressions in algebra is crucial in solving complex problems and understanding mathematical concepts.
• How do I apply the rules of exponents? To apply the rules of exponents, you need to remember the following rules: when multiplying two numbers with the same base, add their exponents; when dividing two numbers with the same base, subtract their exponents; when raising a number to a power, multiply the exponents.

Final Answer

The final answer is $\boxed{\frac{24}{5}}$.