Solve The Following Equation:$49^{x+2}=\left(\frac{1}{7}\right)^{-x-7}$A. $x = 4$ B. $x = 7$ C. $x = -7$ D. $x = 3$

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of algebraic manipulations and properties of exponents. In this article, we will focus on solving a specific exponential equation, $49^{x+2}=\left(\frac{1}{7}\right)^{-x-7}$, and explore the different methods and techniques used to find the solution.

Understanding Exponents

Before we dive into solving the equation, let's review the basics of exponents. An exponent is a small number that is placed above and to the right of a base number, indicating how many times the base number should be multiplied by itself. For example, $2^3$ means $2$ multiplied by itself $3$ times, or $2 \times 2 \times 2 = 8$. Exponents can also be negative, which means the base number is divided by itself a certain number of times.

The Equation

The equation we are given is $49^{x+2}=\left(\frac{1}{7}\right)^{-x-7}$. To solve this equation, we need to isolate the variable $x$ and find its value. Let's start by simplifying the equation using the properties of exponents.

Simplifying the Equation

We can start by rewriting $49$ as $7^2$, since $49$ is equal to $7$ squared. This gives us:

$$\left(7^2\right)^{x+2}=\left(\frac{1}{7}\right)^{-x-7}$$

Using the property of exponents that states $(a^b)^c = a^{bc}$, we can simplify the left-hand side of the equation:

$$7^{2(x+2)}=\left(\frac{1}{7}\right)^{-x-7}$$

Now, let's focus on the right-hand side of the equation. We can rewrite $\frac{1}{7}$ as $7^{-1}$, since $\frac{1}{7}$ is equal to $7$ to the power of $-1$. This gives us:

$$7^{2(x+2)}=7^{-1(-x-7)}$$

Using the property of exponents that states $a^{-b} = \frac{1}{a^b}$, we can simplify the right-hand side of the equation:

$$7^{2(x+2)}=7^{x+7}$$

Equating the Exponents

Now that we have simplified both sides of the equation, we can equate the exponents:

$$2(x+2)=x+7$$

To solve for $x$, we can start by distributing the $2$ on the left-hand side of the equation:

$$2x+4=x+7$$

Next, we can subtract $x$ from both sides of the equation to get:

$$x+4=7$$

Finally, we can subtract $4$ from both sides of the equation to solve for $x$:

$$x=3$$

Conclusion

In this article, we solved the exponential equation $49^{x+2}=\left(\frac{1}{7}\right)^{-x-7}$ using the properties of exponents and algebraic manipulations. We simplified the equation by rewriting $49$ as $7^2$ and using the property of exponents that states $(a^b)^c = a^{bc}$. We then equated the exponents and solved for $x$ to find the solution.

Answer

The solution to the equation is $x = 3$.

Discussion

This equation is a classic example of an exponential equation, and solving it requires a deep understanding of algebraic manipulations and properties of exponents. The solution we found, $x = 3$, is the only possible solution to the equation, and it can be verified by plugging it back into the original equation.

Related Topics

• Exponential equations
• Algebraic manipulations
• Properties of exponents
• Solving equations

Further Reading

• Solving Exponential Equations
• Properties of Exponents
• Algebraic Manipulations

References

• Math Is Fun
• Khan Academy
• Wikipedia
  Solving Exponential Equations: A Q&A Guide =====================================================

Introduction

In our previous article, we solved the exponential equation $49^{x+2}=\left(\frac{1}{7}\right)^{-x-7}$ using the properties of exponents and algebraic manipulations. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques used to solve exponential equations.

Q&A

Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential expression, which is a number raised to a power. For example, $2^3$ is an exponential expression, where $2$ is the base and $3$ is the exponent.

Q: How do I simplify an exponential equation?

A: To simplify an exponential equation, you can use the properties of exponents, such as the product rule, the quotient rule, and the power rule. For example, if you have the equation $2^3 \cdot 2^4$, you can simplify it by using the product rule, which states that $a^m \cdot a^n = a^{m+n}$.

Q: What is the difference between a linear equation and an exponential equation?

A: A linear equation is an equation that involves a linear expression, which is a polynomial of degree one. For example, $2x + 3$ is a linear equation. An exponential equation, on the other hand, involves an exponential expression, which is a number raised to a power. For example, $2^x$ is an exponential equation.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you can use the properties of exponents and algebraic manipulations. For example, if you have the equation $2^x = 8$, you can solve it by using the property of exponents that states $a^m = a^n$ implies $m = n$.

Q: What is the significance of the base and exponent in an exponential equation?

A: The base and exponent in an exponential equation are crucial in determining the solution. The base represents the number being raised to a power, while the exponent represents the power to which the base is being raised. For example, in the equation $2^x = 8$, the base is $2$ and the exponent is $x$. The value of $x$ determines the solution to the equation.

Q: Can I use logarithms to solve exponential equations?

A: Yes, you can use logarithms to solve exponential equations. Logarithms are the inverse operation of exponentiation, and they can be used to solve exponential equations by converting them into linear equations.

Q: What are some common mistakes to avoid when solving exponential equations?

A: Some common mistakes to avoid when solving exponential equations include:

• Not using the properties of exponents correctly
• Not simplifying the equation properly
• Not isolating the variable correctly
• Not checking the solution by plugging it back into the original equation

Conclusion

Solving exponential equations requires a deep understanding of the properties of exponents and algebraic manipulations. By following the steps outlined in this Q&A guide, you can better understand the concepts and techniques used to solve exponential equations. Remember to always check your solution by plugging it back into the original equation to ensure that it is correct.

Related Topics

• Exponential equations
• Algebraic manipulations
• Properties of exponents
• Logarithms
• Solving equations

Further Reading

• Solving Exponential Equations
• Properties of Exponents
• Algebraic Manipulations
• Logarithms

References

• Math Is Fun
• Khan Academy
• Wikipedia