Solve:$\[ \frac{512^{x-2}}{\left(\frac{1}{64}\right)^{3x}} = 512 \\]A. \[$ X = -1 \$\]B. \[$ X = 0 \$\]C. \[$ X = 1 \$\]D. No Solution

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the underlying principles. In this article, we will focus on solving a specific type of exponential equation, which involves fractions and powers of numbers. We will use the given equation as a case study to illustrate the steps involved in solving such equations.

The Given Equation

The given equation is:

$$\frac{512^{x-2}}{\left(\frac{1}{64}\right)^{3x}} = 512$$

Step 1: Simplify the Equation

To simplify the equation, we need to express both sides in terms of the same base. We can rewrite 512 as $2^9$ and $\frac{1}{64}$ as $2^{-3}$.

$$\frac{(2^9)^{x-2}}{(2^{-3})^{3x}} = 2^9$$

Using the property of exponents that $(a^b)^c = a^{bc}$, we can simplify the equation further.

$$\frac{2^{9(x-2)}}{2^{-3(3x)}} = 2^9$$

Step 2: Use the Quotient Rule

The quotient rule states that $\frac{a^b}{a^c} = a^{b-c}$. We can use this rule to simplify the left-hand side of the equation.

$$2^{9(x-2) - (-3)(3x)} = 2^9$$

Simplifying the exponent, we get:

$$2^{9x-18+9x} = 2^9$$

Combine like terms:

$$2^{18x-18} = 2^9$$

Step 3: Equate the Exponents

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

$$18x-18 = 9$$

Add 18 to both sides:

$$18x = 27$$

Divide both sides by 18:

$$x = \frac{27}{18}$$

Simplify the fraction:

$$x = \frac{3}{2}$$

Conclusion

In this article, we solved a specific type of exponential equation that involved fractions and powers of numbers. We used the quotient rule and the property of exponents to simplify the equation and eventually equate the exponents. The solution to the equation is $x = \frac{3}{2}$.

Discussion

The given equation is a classic example of an exponential equation that requires careful simplification and manipulation. The use of the quotient rule and the property of exponents is essential in solving such equations. The solution to the equation is a rational number, which is a common outcome in exponential equations.

Comparison with Options

The solution to the equation is $x = \frac{3}{2}$, which is not among the given options. Therefore, the correct answer is:

No solution

Final Answer

$$

Introduction

In our previous article, we solved a specific type of exponential equation that involved fractions and powers of numbers. In this article, we will provide a Q&A guide to help you understand the concepts and techniques involved in solving exponential equations.

Q: What is an exponential equation?

A: An exponential equation is an equation that involves a variable in the exponent of a number. For example, the equation $2^x = 8$ is an exponential equation.

Q: How do I simplify an exponential equation?

A: To simplify an exponential equation, you need to express both sides in terms of the same base. You can use the property of exponents that $(a^b)^c = a^{bc}$ to simplify the equation.

Q: What is the quotient rule for exponents?

A: The quotient rule for exponents states that $\frac{a^b}{a^c} = a^{b-c}$. This rule can be used to simplify the left-hand side of an exponential equation.

Q: How do I equate the exponents in an exponential equation?

A: Since the bases are the same, you can equate the exponents. For example, in the equation $2^{18x-18} = 2^9$, you can equate the exponents to get $18x-18 = 9$.

Q: What is the solution to the given equation?

A: The solution to the given equation is $x = \frac{3}{2}$.

Q: Why is the solution not among the given options?

A: The solution to the equation is a rational number, which is not among the given options.

Q: What is the correct answer?

A: The correct answer is No solution.

Q: How do I apply the concepts and techniques to other exponential equations?

A: To apply the concepts and techniques to other exponential equations, you need to:

1. Simplify the equation by expressing both sides in terms of the same base.
2. Use the quotient rule to simplify the left-hand side of the equation.
3. Equate the exponents to solve for the variable.

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

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

1. Not expressing both sides in terms of the same base.
2. Not using the quotient rule to simplify the left-hand side of the equation.
3. Not equating the exponents to solve for the variable.

Conclusion

In this article, we provided a Q&A guide to help you understand the concepts and techniques involved in solving exponential equations. We covered topics such as simplifying exponential equations, using the quotient rule, and equating exponents. We also discussed common mistakes to avoid when solving exponential equations.

Final Answer

The final answer is $\boxed{\frac{3}{2}}$.