If 343 X = 49 4 − X 343^x = 49^{4-x} 34 3 X = 4 9 4 − X , What Is The Value Of X X X ?A. 5 8 \frac{5}{8} 8 5 ​ B. 8 5 \frac{8}{5} 5 8 ​ C. 2 D. 8

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Introduction

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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 type of exponential equation, namely, the equation $343^x = 49^{4-x}$. Our goal is to find the value of $x$ that satisfies this equation.

Understanding Exponents

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Before we dive into solving the equation, let's take a moment to understand the concept of exponents. An exponent is a small number that is raised to a power, indicating how many times a base number is multiplied by itself. For example, $2^3$ means $2$ multiplied by itself $3$ times, which equals $8$. In general, $a^b$ means $a$ multiplied by itself $b$ times.

Properties of Exponents

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Exponents have several important properties that we will use to solve the equation. These properties include:

• Product of Powers: When multiplying two numbers with the same base, we add their exponents. For example, $a^m \cdot a^n = a^{m+n}$.
• Power of a Power: When raising a number with an exponent to another power, we multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$.
• Zero Exponent: Any non-zero number raised to the power of zero is equal to $1$. For example, $a^0 = 1$.

Solving the Equation

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Now that we have a good understanding of exponents and their properties, let's solve the equation $343^x = 49^{4-x}$. To do this, we will use the properties of exponents to simplify the equation and isolate the variable $x$.

Step 1: Simplify the Equation

First, let's simplify the equation by expressing both sides with the same base. We know that $343 = 7^3$ and $49 = 7^2$. Therefore, we can rewrite the equation as:

$(7^3)^x = (7^2)^{4-x}$

Using the property of power of a power, we can simplify the equation further:

$7^{3x} = 7^{2(4-x)}$

Step 2: Equate the Exponents

Since the bases are the same, we can equate the exponents:

$3x = 2(4-x)$

Step 3: Solve for $x$

Now, let's solve for $x$ by simplifying the equation:

$3x = 8 - 2x$

Add $2x$ to both sides:

$5x = 8$

Divide both sides by $5$:

$x = \frac{8}{5}$

Conclusion

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In this article, we solved the exponential equation $343^x = 49^{4-x}$ by using the properties of exponents and simplifying the equation. We found that the value of $x$ that satisfies the equation is $\frac{8}{5}$. This solution demonstrates the importance of understanding exponents and their properties in solving mathematical equations.

Final Answer

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The final answer is $\boxed{\frac{8}{5}}$.

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Introduction

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In our previous article, we solved the exponential equation $343^x = 49^{4-x}$ and found that the value of $x$ that satisfies the equation is $\frac{8}{5}$. However, we understand that readers may have questions about the solution and the process of solving exponential equations. In this article, we will address some of the most frequently asked questions about solving exponential equations.

Q&A

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

A: An exponential equation is an equation that involves an exponential expression, such as $a^x = b$, where $a$ and $b$ are constants and $x$ is the variable. A linear equation, on the other hand, is an equation that involves a linear expression, such as $ax + b = c$, where $a$, $b$, and $c$ are constants.

Q: How do I know which property of exponents to use when solving an exponential equation?

A: When solving an exponential equation, you need to use the properties of exponents to simplify the equation and isolate the variable. The properties of exponents include the product of powers, power of a power, and zero exponent. You need to choose the property that is most relevant to the equation you are solving.

Q: Can I use the same method to solve all exponential equations?

A: No, you cannot use the same method to solve all exponential equations. The method you use will depend on the specific equation you are solving and the properties of exponents that are involved. For example, if the equation involves a base with a fractional exponent, you may need to use a different method to solve it.

Q: How do I know if an exponential equation has a solution?

A: An exponential equation may or may not have a solution, depending on the values of the constants and the variable. If the equation is inconsistent, it will not have a solution. If the equation is consistent, it will have a solution, but it may be a complex number or an irrational number.

Q: Can I use a calculator to solve exponential equations?

A: Yes, you can use a calculator to solve exponential equations, but you need to be careful when using a calculator. A calculator can only give you an approximate solution, and it may not be accurate. It's always best to use a calculator as a check on your solution, rather than relying on it to give you the exact solution.

Q: How do I check my solution to an exponential equation?

A: To check your solution to an exponential equation, you need to plug it back into the original equation and simplify it. If the simplified expression is equal to the original expression, then your solution is correct. If the simplified expression is not equal to the original expression, then your solution is incorrect.

Conclusion

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In this article, we addressed some of the most frequently asked questions about solving exponential equations. We hope that this article has been helpful in clarifying any confusion you may have had about solving exponential equations. Remember to always use the properties of exponents to simplify the equation and isolate the variable, and to check your solution to ensure that it is correct.

Final Tips

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• Always read the problem carefully and understand what is being asked.
• Use the properties of exponents to simplify the equation and isolate the variable.
• Check your solution to ensure that it is correct.
• Use a calculator as a check on your solution, rather than relying on it to give you the exact solution.

Additional Resources

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• Khan Academy: Exponents and Exponential Functions
• Mathway: Exponential Equations
• Wolfram Alpha: Exponential Equations

We hope that this article has been helpful in providing you with a better understanding of solving exponential equations. If you have any further questions or need additional help, please don't hesitate to ask.