Explain The Error In The Work Shown. Find The Correct Answer.${ \begin{array}{l} \frac{1}{64}=16^{2a} \ \left(2 4\right) {-3}=\left(2 4\right) {2a} \ 2 {-12}=2 {8a} \ -12=8a \ a=-\frac{12}{8} \ a=-\frac{3}{2} \end{array} }$

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 analyze a given work that attempts to solve an exponential equation and identify the errors made in the process. We will then provide the correct solution to the equation.

The Given Work

The given work is as follows:

{ \begin{array}{l} \frac{1}{64}=16^{2a} \\ \left(2^4\right)^{-3}=\left(2^4\right)^{2a} \\ 2^{-12}=2^{8a} \\ -12=8a \\ a=-\frac{12}{8} \\ a=-\frac{3}{2} \end{array} \}

Error 1: Incorrect Application of Exponent Rules

The first error in the given work is the incorrect application of exponent rules. In the first equation, $\frac{1}{64}=16^{2a}$, the left-hand side is a fraction, while the right-hand side is an exponential expression. To make the equation true, the base of the exponential expression should be the same as the base of the fraction. However, the base of the fraction is 64, which is equal to $4^3$, not 16.

Correcting the Error

To correct the error, we should rewrite the first equation as follows:

$\frac{1}{64}=\left(4^3\right)^{2a}$

Using the property of exponents that $(a^m)^n=a^{mn}$, we can rewrite the equation as:

$\frac{1}{64}=4^{6a}$

Error 2: Incorrect Simplification

The second error in the given work is the incorrect simplification of the second equation. In the second equation, $\left(2^4\right)^{-3}=\left(2^4\right)^{2a}$, the left-hand side is equal to $2^{-12}$, while the right-hand side is equal to $2^{8a}$. To make the equation true, the exponents on both sides should be equal.

Correcting the Error

To correct the error, we should rewrite the second equation as follows:

$2^{-12}=2^{8a}$

Using the property of exponents that $a^m=a^n$ implies $m=n$, we can conclude that:

$-12=8a$

Error 3: Incorrect Solution

The third error in the given work is the incorrect solution to the equation. In the given work, the solution to the equation is $a=-\frac{3}{2}$. However, this solution is incorrect because it does not satisfy the original equation.

Correcting the Error

To correct the error, we should solve the equation $-12=8a$ correctly. Dividing both sides of the equation by 8, we get:

$a=-\frac{12}{8}$

Simplifying the fraction, we get:

$a=-\frac{3}{2}$

However, this solution is still incorrect because it does not satisfy the original equation. To find the correct solution, we should substitute the value of $a$ back into the original equation and check if it is true.

Correct Solution

Substituting $a=-\frac{3}{2}$ back into the original equation, we get:

$\frac{1}{64}=\left(4^3\right)^{-3}$

Using the property of exponents that $(a^m)^n=a^{mn}$, we can rewrite the equation as:

$\frac{1}{64}=4^{-9}$

Using the property of exponents that $a^{-m}=\frac{1}{a^m}$, we can rewrite the equation as:

$\frac{1}{64}=\frac{1}{4^9}$

Since $\frac{1}{64}=\frac{1}{4^3}$, we can conclude that:

$4^3=4^9$

This is a true statement, and therefore, the solution $a=-\frac{3}{2}$ is correct.

Conclusion

In conclusion, the given work contains three errors: incorrect application of exponent rules, incorrect simplification, and incorrect solution. We have corrected these errors and provided the correct solution to the equation. The correct solution is $a=-\frac{3}{2}$, which satisfies the original equation.

Final Answer

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Introduction

In our previous article, we analyzed a given work that attempted to solve an exponential equation and identified the errors made in the process. We then provided the correct solution to the equation. In this article, we will answer some frequently asked questions related to exponential equations and error analysis.

Q: What are exponential equations?

A: Exponential equations are equations that involve exponential expressions, which are expressions of the form $a^b$, where $a$ is the base and $b$ is the exponent.

Q: What are some common errors made when solving exponential equations?

A: Some common errors made when solving exponential equations include:

• Incorrect application of exponent rules
• Incorrect simplification of expressions
• Incorrect solution to the equation

Q: How can I avoid making these errors?

A: To avoid making these errors, you should:

• Carefully read and understand the equation
• Apply exponent rules correctly
• Simplify expressions correctly
• Check your solution to ensure it satisfies the original equation

Q: What is the correct solution to the equation $\frac{1}{64}=16^{2a}$?

A: The correct solution to the equation $\frac{1}{64}=16^{2a}$ is $a=-\frac{3}{2}$.

Q: How do I know if my solution is correct?

A: To ensure your solution is correct, you should:

• Substitute your solution back into the original equation
• Check if the equation is true
• Verify that your solution satisfies the original equation

Q: What are some tips for solving exponential equations?

A: Some tips for solving exponential equations include:

• Use exponent rules to simplify expressions
• Use properties of exponents to rewrite expressions
• Check your solution to ensure it satisfies the original equation

Q: Can you provide an example of an exponential equation?

A: Here is an example of an exponential equation:

$2^x=8$

To solve this equation, you would need to:

• Use exponent rules to rewrite the equation
• Use properties of exponents to simplify the expression
• Check your solution to ensure it satisfies the original equation

Q: How do I know if an exponential equation is true or false?

A: To determine if an exponential equation is true or false, you should:

• Substitute values into the equation
• Check if the equation is true
• Verify that the equation satisfies the original equation

Conclusion

In conclusion, exponential equations can be challenging to solve, but with practice and patience, you can become proficient in solving them. By understanding the common errors made when solving exponential equations and following the tips provided, you can ensure that your solutions are correct.

Final Answer

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