Solving The Exponential Equation 512^(x-2) / (1/64)^(3x) = 512

This comprehensive guide provides a detailed solution to the exponential equation $\frac{512^{x-2}}{\left(\frac{1}{64}\right)^{3 x}}=512$. Exponential equations, where the variable appears in the exponent, can seem daunting at first. However, by applying the fundamental principles of exponents and logarithms, we can systematically solve these equations. This article aims to provide a clear, step-by-step solution, making the process accessible to anyone with a basic understanding of algebra. We'll explore the properties of exponents, demonstrate how to manipulate the equation, and arrive at the correct solution. Understanding how to solve exponential equations is crucial in various fields, including mathematics, physics, engineering, and finance, where exponential models are frequently used to describe growth, decay, and other phenomena. By mastering these techniques, you'll be well-equipped to tackle a wide range of problems involving exponential relationships.

Problem Statement

The given exponential equation is:

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

Our objective is to find the value of x that satisfies this equation. To do this, we will leverage the properties of exponents to simplify the equation and isolate the variable x. Exponential equations often require manipulation to bring them into a form where the bases are the same, allowing us to equate the exponents. This problem is a classic example of how we can use exponent rules to simplify complex expressions and find solutions. We will begin by expressing all the terms in the equation with the same base, which is a common strategy when dealing with exponential equations. This will allow us to combine the exponents and ultimately solve for x. The process involves understanding the relationships between numbers like 512 and 64, which are powers of 2, and applying the rules of exponents to simplify the expression.

Step-by-Step Solution

Here’s a detailed breakdown of how to solve the equation:

1. Express all terms with the same base

The first key step in solving this exponential equation is to express all terms with the same base. This allows us to directly compare the exponents and simplify the equation. We observe that 512 and 64 can both be expressed as powers of 2. Specifically, $512 = 2^9$ and $64 = 2^6$. Additionally, $\frac{1}{64}$ can be written as $2^{-6}$. This conversion is crucial because it allows us to rewrite the entire equation in terms of a single base, which simplifies the subsequent steps. Expressing numbers as powers of a common base is a fundamental technique in solving exponential equations and inequalities. Once we have a common base, we can use the properties of exponents to combine terms and solve for the unknown variable. In this case, identifying that 512 and 64 are powers of 2 is the starting point for simplifying the given equation and making it easier to manipulate.

Substituting these values into the original equation, we get:

$$\frac{(2^9)^{x-2}}{(2^{-6})^{3x}} = 2^9$$

2. Apply the power of a power rule

Next, we apply the power of a power rule, which states that $(a^m)^n = a^{mn}$. This rule is fundamental in simplifying expressions where an exponent is raised to another exponent. By applying this rule, we can eliminate the parentheses and combine the exponents. Understanding and applying this rule correctly is crucial for solving exponential equations and other mathematical problems involving exponents. It allows us to reduce complex expressions to simpler forms, making them easier to manipulate and solve. In this context, we'll apply the rule to both the numerator and the denominator of the left side of the equation, which will help us further simplify the expression and move closer to isolating the variable x.

Applying this rule, the equation becomes:

$$\frac{2^{9(x-2)}}{2^{-18x}} = 2^9$$

3. Simplify the exponents

Now, let's simplify the exponents in the equation. Multiplying the exponents in the numerator, we get $9(x-2) = 9x - 18$. This simplification is a direct application of the distributive property of multiplication over subtraction. Similarly, in the denominator, the exponent remains as $-18x$. Simplifying exponents is a critical step in solving exponential equations, as it helps in reducing the complexity of the equation and making it easier to manipulate. By performing these simplifications, we can combine terms with the same base and eventually solve for the unknown variable. In this case, simplifying the exponents sets the stage for using the quotient rule of exponents, which will further consolidate the terms on the left side of the equation.

Thus, the equation is now:

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

4. Apply the quotient rule of exponents

To further simplify the equation, we apply the quotient rule of exponents. This rule states that $\frac{a^m}{a^n} = a^{m-n}$. This is a fundamental property of exponents that allows us to simplify expressions involving the division of powers with the same base. By subtracting the exponents in the denominator from the exponents in the numerator, we can combine the terms into a single exponential expression. This step is essential in reducing the equation to a simpler form, making it easier to solve for the unknown variable. Understanding and correctly applying the quotient rule is crucial for manipulating exponential expressions and solving exponential equations efficiently.

Applying this rule, we subtract the exponent in the denominator from the exponent in the numerator:

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

5. Combine like terms in the exponent

Next, we combine like terms in the exponent. Combining like terms is a fundamental algebraic technique used to simplify expressions by grouping and adding or subtracting terms that contain the same variable or constant. In this context, we have the expression $(9x - 18) - (-18x)$ in the exponent. To simplify this, we need to remove the parentheses and combine the x terms and the constant terms separately. This step is crucial in reducing the complexity of the equation and making it easier to solve for the unknown variable x. Correctly combining like terms ensures that the equation is accurately simplified, which is essential for obtaining the correct solution. This process sets the stage for equating the exponents in the next step, which will allow us to solve for x directly.

Simplifying the exponent, we get:

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

$$2^{27x-18} = 2^9$$

6. Equate the exponents

Now that we have the same base on both sides of the equation, we can equate the exponents. This is a critical step in solving exponential equations when the bases are the same. The principle behind this step is that if $a^m = a^n$, then $m = n$. By equating the exponents, we transform the exponential equation into a linear equation, which is much easier to solve. This step significantly simplifies the problem, allowing us to isolate the variable x and find its value. It is essential to ensure that the bases are indeed the same before equating the exponents, as this is a prerequisite for the validity of this step. Once we equate the exponents, we can use basic algebraic techniques to solve for x.

Equating the exponents, we have:

$$27x - 18 = 9$$

7. Solve for x

To solve for x, we need to isolate it on one side of the equation. This involves performing algebraic operations to both sides of the equation in order to isolate the variable. First, we add 18 to both sides of the equation, which cancels out the -18 term on the left side. Then, we divide both sides by 27 to solve for x. Each of these steps maintains the equality of the equation while bringing us closer to the solution. Solving for a variable is a fundamental skill in algebra and is essential for solving a wide range of mathematical problems. In this context, accurately solving for x will give us the value that satisfies the original exponential equation.

Adding 18 to both sides:

$$27x = 9 + 18$$

$$27x = 27$$

Dividing by 27:

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

$$x = 1$$

Final Answer

Therefore, the solution to the equation is:

$$x = 1$$

Answer Choice

Thus, the correct answer is:

C. $x = 1$

Conclusion

In conclusion, solving the exponential equation $\frac{512^{x-2}}{\left(\frac{1}{64}\right)^{3 x}}=512$ involves several key steps, primarily focusing on expressing all terms with a common base and applying the properties of exponents. By converting 512 and 64 into powers of 2, we simplified the equation and made it easier to manipulate. The application of the power of a power rule and the quotient rule of exponents was crucial in reducing the complexity of the equation. Once we had the same base on both sides, equating the exponents allowed us to transform the exponential equation into a simple linear equation, which we then solved for x. This step-by-step process demonstrates how exponential equations can be systematically solved by understanding and applying the fundamental principles of exponents. The final solution, $x = 1$, satisfies the original equation, confirming the accuracy of our approach. This method is applicable to a wide range of exponential equations, making it a valuable tool in mathematical problem-solving.