Solve The Equation:${ 4^2 \cdot 64^{2x+1} = \left(\frac{1}{16}\right)^x }$

$$

Introduction

In this article, we will delve into solving a complex equation involving exponents and powers. The equation given is $4^2 \cdot 64^{2x+1} = \left(\frac{1}{16}\right)^x$. We will break down the equation step by step, using mathematical concepts and techniques to simplify and solve for the variable $x$. Our goal is to provide a clear and concise solution to this equation, making it accessible to readers with a basic understanding of algebra and exponent rules.

Understanding the Equation

The given equation involves three main components: $4^2$, $64^{2x+1}$, and $\left(\frac{1}{16}\right)^x$. To begin solving the equation, we need to understand the properties of exponents and how they interact with each other.

• Exponent Rules: When dealing with exponents, it's essential to remember the following rules:
  • $a^m \cdot a^n = a^{m+n}$
  • $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$
  • $(a^m)^n = a^{m \cdot n}$

Simplifying the Equation

To simplify the equation, we can start by expressing $64$ as a power of $4$, since $64 = 4^3$. This allows us to rewrite the equation as:

$$4^2 \cdot (4^3)^{2x+1} = \left(\frac{1}{16}\right)^x$$

Using the exponent rule $(a^m)^n = a^{m \cdot n}$, we can simplify the left-hand side of the equation:

$$4^2 \cdot 4^{3(2x+1)} = \left(\frac{1}{16}\right)^x$$

Applying Exponent Rules

Now that we have simplified the equation, we can apply the exponent rule $a^m \cdot a^n = a^{m+n}$ to combine the exponents on the left-hand side:

$$4^{2+3(2x+1)} = \left(\frac{1}{16}\right)^x$$

Simplifying the Exponents

To simplify the exponents, we can start by evaluating the expression $3(2x+1)$:

$$3(2x+1) = 6x + 3$$

Now, we can substitute this expression back into the equation:

$$4^{2+6x+3} = \left(\frac{1}{16}\right)^x$$

Simplifying the Exponents (Continued)

We can further simplify the exponents by combining the constants:

$$4^{6x+5} = \left(\frac{1}{16}\right)^x$$

Expressing $\left(\frac{1}{16}\right)^x$ as a Power of 4

To make it easier to compare the two sides of the equation, we can express $\left(\frac{1}{16}\right)^x$ as a power of $4$. Since $\frac{1}{16} = 4^{-2}$, we can rewrite the equation as:

$$4^{6x+5} = (4^{-2})^x$$

Applying Exponent Rules (Again)

Using the exponent rule $(a^m)^n = a^{m \cdot n}$, we can simplify the right-hand side of the equation:

$$4^{6x+5} = 4^{-2x}$$

Equating the Exponents

Since the bases are the same ($4$), we can equate the exponents:

$$6x + 5 = -2x$$

Solving for $x$

To solve for $x$, we can add $2x$ to both sides of the equation:

$$8x + 5 = 0$$

Next, we can subtract $5$ from both sides:

$$8x = -5$$

Finally, we can divide both sides by $8$:

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

Conclusion

In this article, we have solved the equation $4^2 \cdot 64^{2x+1} = \left(\frac{1}{16}\right)^x$ using mathematical concepts and techniques. We have broken down the equation step by step, simplifying and solving for the variable $x$. The final solution is $x = -\frac{5}{8}$.

Final Thoughts

Solving equations involving exponents and powers requires a deep understanding of mathematical concepts and techniques. By following the steps outlined in this article, readers can develop their problem-solving skills and become more confident in their ability to tackle complex mathematical equations.

Additional Resources

For readers who want to learn more about exponents and powers, we recommend checking out the following resources:

• Khan Academy: Exponents and Powers
• Mathway: Exponents and Powers
• Wolfram Alpha: Exponents and Powers

By exploring these resources, readers can gain a deeper understanding of mathematical concepts and techniques, and become more proficient in solving complex equations.

$$

Introduction

In our previous article, we solved the equation $4^2 \cdot 64^{2x+1} = \left(\frac{1}{16}\right)^x$ using mathematical concepts and techniques. In this article, we will provide a Q&A section to help readers understand the solution and address any questions they may have.

Q&A

Q: What is the final solution to the equation?

A: The final solution to the equation is $x = -\frac{5}{8}$.

Q: How did you simplify the equation?

A: We simplified the equation by expressing $64$ as a power of $4$, using exponent rules to combine the exponents, and equating the exponents.

Q: What is the significance of the exponent rule $(a^m)^n = a^{m \cdot n}$?

A: The exponent rule $(a^m)^n = a^{m \cdot n}$ allows us to simplify expressions involving exponents and powers. It states that when we raise a power to another power, we can multiply the exponents.

Q: How did you express $\left(\frac{1}{16}\right)^x$ as a power of $4$?

A: We expressed $\left(\frac{1}{16}\right)^x$ as a power of $4$ by using the fact that $\frac{1}{16} = 4^{-2}$. This allowed us to rewrite the equation as $4^{6x+5} = (4^{-2})^x$.

Q: What is the difference between $4^{6x+5}$ and $4^{-2x}$?

A: The difference between $4^{6x+5}$ and $4^{-2x}$ is the exponent. $4^{6x+5}$ has a positive exponent, while $4^{-2x}$ has a negative exponent.

Q: How did you solve for $x$?

A: We solved for $x$ by equating the exponents, adding $2x$ to both sides of the equation, subtracting $5$ from both sides, and dividing both sides by $8$.

Q: What is the importance of understanding exponent rules?

A: Understanding exponent rules is crucial in solving equations involving exponents and powers. It allows us to simplify expressions, equate exponents, and solve for variables.

Conclusion

In this Q&A article, we have addressed common questions and concerns related to solving the equation $4^2 \cdot 64^{2x+1} = \left(\frac{1}{16}\right)^x$. We hope that this article has provided readers with a deeper understanding of the solution and has helped them to develop their problem-solving skills.

Final Thoughts

Solving equations involving exponents and powers requires a deep understanding of mathematical concepts and techniques. By following the steps outlined in this article and practicing with different problems, readers can become more confident in their ability to tackle complex mathematical equations.

Additional Resources

For readers who want to learn more about exponents and powers, we recommend checking out the following resources:

• Khan Academy: Exponents and Powers
• Mathway: Exponents and Powers
• Wolfram Alpha: Exponents and Powers

By exploring these resources, readers can gain a deeper understanding of mathematical concepts and techniques, and become more proficient in solving complex equations.

Common Mistakes to Avoid

When solving equations involving exponents and powers, it's essential to avoid common mistakes. Here are a few to watch out for:

• Not simplifying the equation: Failing to simplify the equation can lead to incorrect solutions.
• Not equating the exponents: Failing to equate the exponents can result in incorrect solutions.
• Not following the order of operations: Failing to follow the order of operations (PEMDAS) can lead to incorrect solutions.

By avoiding these common mistakes, readers can ensure that they are solving equations correctly and accurately.

Practice Problems

To practice solving equations involving exponents and powers, we recommend trying the following problems:

• $2^3 \cdot 8^{2x+1} = \left(\frac{1}{4}\right)^x$
• $3^2 \cdot 27^{2x+1} = \left(\frac{1}{9}\right)^x$
• $4^2 \cdot 64^{2x+1} = \left(\frac{1}{16}\right)^x$

By practicing with these problems, readers can develop their problem-solving skills and become more confident in their ability to tackle complex mathematical equations.