(Simplify): 49 X + 7 2 X − 1 8 × 7 2 X − 2 \frac{49^x+7^{2x-1}}{8 \times 7^{2x-2}} 8 × 7 2 X − 2 4 9 X + 7 2 X − 1 ​

Introduction

In this article, we will delve into the world of algebra and simplify a complex expression involving exponents. The given expression is $\frac{49^x+7^{2x-1}}{8 \times 7^{2x-2}}$. Our goal is to simplify this expression and make it more manageable. We will use various algebraic techniques, including exponent rules and factoring, to achieve this goal.

Understanding the Expression

Before we begin simplifying the expression, let's break it down and understand its components. The numerator of the expression is $49^x+7^{2x-1}$, while the denominator is $8 \times 7^{2x-2}$. We can see that both terms in the numerator have a base of 7, but with different exponents. Similarly, the denominator has a base of 7, but with a different exponent.

Simplifying the Numerator

Let's start by simplifying the numerator. We can rewrite $49^x$ as $(7^2)^x$, which is equal to $7^{2x}$. Now, we have $7^{2x}+7^{2x-1}$ in the numerator. We can factor out a common term of $7^{2x-1}$ from both terms, resulting in $7^{2x-1}(7+1)$.

Simplifying the Denominator

Now, let's simplify the denominator. We can rewrite $8 \times 7^{2x-2}$ as $2^3 \times 7^{2x-2}$, which is equal to $2^3 \times 7^{2x-2}$. We can factor out a common term of $7^{2x-3}$ from both terms, resulting in $2^3 \times 7^{2x-3}$.

Combining the Simplified Numerator and Denominator

Now that we have simplified the numerator and denominator, we can combine them to get the final simplified expression. We have $\frac{7^{2x-1}(7+1)}{2^3 \times 7^{2x-3}}$. We can cancel out a common term of $7^{2x-3}$ from both the numerator and denominator, resulting in $\frac{7^{2x-1}(7+1)}{2^3}$.

Further Simplification

We can further simplify the expression by evaluating the term $(7+1)$ in the numerator. This is equal to 8, so we have $\frac{7^{2x-1} \times 8}{2^3}$.

Final Simplification

Now, we can simplify the expression by evaluating the exponent $2x-1$ in the numerator. We can rewrite $7^{2x-1}$ as $\frac{7^{2x}}{7}$, which is equal to $\frac{49^x}{7}$. Now, we have $\frac{\frac{49^x}{7} \times 8}{2^3}$.

Simplifying the Final Expression

We can simplify the final expression by canceling out a common term of 7 from the numerator and denominator, resulting in $\frac{49^x \times 8}{7 \times 2^3}$.

Final Answer

The final simplified expression is $\frac{49^x \times 8}{7 \times 2^3}$.

Conclusion

In this article, we simplified a complex algebraic expression involving exponents. We used various algebraic techniques, including exponent rules and factoring, to achieve this goal. The final simplified expression is $\frac{49^x \times 8}{7 \times 2^3}$.

Future Work

In future work, we can explore other algebraic expressions and simplify them using various techniques. We can also explore the properties of exponents and how they can be used to simplify complex expressions.

References

• [1] Algebraic Expressions and Equations, by Michael Artin
• [2] Exponents and Logarithms, by James Stewart

Keywords

• Algebraic expression
• Exponents
• Simplification
• Factoring
• Algebra
  

Introduction

In our previous article, we simplified the algebraic expression $\frac{49^x+7^{2x-1}}{8 \times 7^{2x-2}}$. In this article, we will answer some frequently asked questions (FAQs) related to the simplification of this expression.

Q: What is the final simplified expression?

A: The final simplified expression is $\frac{49^x \times 8}{7 \times 2^3}$.

Q: How did you simplify the numerator?

A: We simplified the numerator by rewriting $49^x$ as $(7^2)^x$, which is equal to $7^{2x}$. We then factored out a common term of $7^{2x-1}$ from both terms, resulting in $7^{2x-1}(7+1)$.

Q: How did you simplify the denominator?

A: We simplified the denominator by rewriting $8 \times 7^{2x-2}$ as $2^3 \times 7^{2x-2}$. We then factored out a common term of $7^{2x-3}$ from both terms, resulting in $2^3 \times 7^{2x-3}$.

Q: Can you explain the concept of exponents in more detail?

A: Exponents are a way of representing repeated multiplication. For example, $7^2$ means $7 \times 7$, and $7^3$ means $7 \times 7 \times 7$. Exponents can be used to simplify complex expressions and make them easier to work with.

Q: How do you handle negative exponents?

A: Negative exponents can be handled by rewriting them as fractions. For example, $7^{-2}$ can be rewritten as $\frac{1}{7^2}$.

Q: Can you provide more examples of simplifying algebraic expressions?

A: Yes, here are a few more examples:

• $\frac{2^x+2^{x-1}}{2 \times 2^{x-1}}$ can be simplified to $\frac{2^x+2^{x-1}}{2 \times 2^{x-1}} = \frac{2^{x-1}(2+1)}{2 \times 2^{x-1}} = \frac{2+1}{2} = \frac{3}{2}$
• $\frac{3^x+3^{x-1}}{3 \times 3^{x-1}}$ can be simplified to $\frac{3^x+3^{x-1}}{3 \times 3^{x-1}} = \frac{3^{x-1}(3+1)}{3 \times 3^{x-1}} = \frac{3+1}{3} = \frac{4}{3}$

Q: How do you know when to simplify an algebraic expression?

A: You should simplify an algebraic expression when it is necessary to make the expression easier to work with. This can be the case when the expression is complex or when you need to perform operations on the expression.

Q: Can you provide more resources for learning algebra?

A: Yes, here are a few resources:

• [1] Algebraic Expressions and Equations, by Michael Artin
• [2] Exponents and Logarithms, by James Stewart
• [3] Khan Academy's Algebra Course
• [4] Mathway's Algebra Calculator

Conclusion

In this article, we answered some frequently asked questions related to the simplification of the algebraic expression $\frac{49^x+7^{2x-1}}{8 \times 7^{2x-2}}$. We provided examples of simplifying algebraic expressions and resources for learning algebra.