Solve For $x$ If $\left(4^{(x \div 3)}\right)\left(16 X\right)=8 {3x}$.

Introduction

Exponential equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will focus on solving a specific exponential equation involving powers of 4, 16, and 8. We will break down the solution into manageable steps, making it easy to understand and follow along.

The Given Equation

The given equation is:

$$\left(4^{(x \div 3)}\right)\left(16^x\right)=8^{3x}$$

Our goal is to solve for the variable $x$.

Step 1: Simplify the Left-Hand Side

To simplify the left-hand side of the equation, we can express 16 as a power of 4:

$$16 = 4^2$$

Substituting this into the equation, we get:

$$\left(4^{(x \div 3)}\right)\left((4^2)^x\right)=8^{3x}$$

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

$$\left(4^{(x \div 3)}\right)\left(4^{2x}\right)=8^{3x}$$

Combining the two terms on the left-hand side, we get:

$$4^{(x \div 3) + 2x} = 8^{3x}$$

Step 2: Express Both Sides with the Same Base

To make it easier to compare the two sides, we can express both sides with the same base. We can rewrite 8 as a power of 2:

$$8 = 2^3$$

Substituting this into the equation, we get:

$$4^{(x \div 3) + 2x} = (2^3)^{3x}$$

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

$$4^{(x \div 3) + 2x} = 2^{9x}$$

Step 3: Equate the Exponents

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

$$(x \div 3) + 2x = 9x$$

To solve for $x$, we can start by multiplying both sides by 3 to eliminate the fraction:

$$x + 6x = 27x$$

Combining like terms, we get:

$$7x = 27x$$

Subtracting $7x$ from both sides, we get:

$$0 = 20x$$

Dividing both sides by 20, we get:

$$x = 0$$

Conclusion

In this article, we solved an exponential equation involving powers of 4, 16, and 8. We broke down the solution into manageable steps, making it easy to understand and follow along. By simplifying the left-hand side, expressing both sides with the same base, and equating the exponents, we were able to solve for the variable $x$. The final solution is $x = 0$.

Additional Tips and Tricks

• When solving exponential equations, it's essential to simplify the left-hand side and express both sides with the same base.
• Use the properties of exponents to simplify and manipulate the equation.
• Equate the exponents when the bases are the same.
• Be careful when multiplying and dividing fractions.

Common Mistakes to Avoid

• Failing to simplify the left-hand side of the equation.
• Not expressing both sides with the same base.
• Not equating the exponents when the bases are the same.
• Making errors when multiplying and dividing fractions.

Real-World Applications

Exponential equations have numerous real-world applications, including:

• Population growth and decline
• Financial modeling and forecasting
• Chemical reactions and kinetics
• Electrical engineering and circuit analysis

By understanding how to solve exponential equations, you can apply these concepts to real-world problems and make informed decisions.

Final Thoughts

Introduction

Exponential equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will address some of the most frequently asked questions about exponential equations, providing clear and concise answers to help you better understand and solve these types of 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 because the variable $x$ is in the exponent of the number 2.

Q: How do I simplify an exponential equation?

A: To simplify an exponential equation, you can start by expressing both sides with the same base. For example, if you have the equation $2^x = 4^y$, you can rewrite 4 as $2^2$, resulting in $2^x = (2^2)^y$. You can then use the property of exponents that $(a^b)^c = a^{bc}$ to simplify further.

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

A: An exponential equation involves a variable in the exponent of a number, while a logarithmic equation involves a variable as the exponent of a number. For example, the equation $2^x = 8$ is an exponential equation, while the equation $x = \log_2 8$ is a logarithmic equation.

Q: How do I solve an exponential equation with a fraction?

A: To solve an exponential equation with a fraction, you can start by simplifying the fraction and then using the properties of exponents to manipulate the equation. For example, if you have the equation $2^{(x \div 3)} = 8$, you can rewrite 8 as $2^3$ and then use the property of exponents that $(a^b)^c = a^{bc}$ to simplify further.

Q: Can I use a calculator to solve an exponential equation?

A: Yes, you can use a calculator to solve an exponential equation. However, it's essential to understand the underlying math and be able to verify the solution using algebraic methods.

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

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

• Failing to simplify the left-hand side of the equation
• Not expressing both sides with the same base
• Not equating the exponents when the bases are the same
• Making errors when multiplying and dividing fractions

Q: How do I apply exponential equations to real-world problems?

A: Exponential equations have numerous real-world applications, including:

• Population growth and decline
• Financial modeling and forecasting
• Chemical reactions and kinetics
• Electrical engineering and circuit analysis

By understanding how to solve exponential equations, you can apply these concepts to real-world problems and make informed decisions.

Q: What are some additional resources for learning about exponential equations?

A: Some additional resources for learning about exponential equations include:

• Online tutorials and videos
• Textbooks and workbooks
• Online communities and forums
• Calculators and software

Conclusion

Exponential equations can be challenging to solve, but with the right approach, they can be tackled with ease. By understanding the basics of exponential equations and practicing with real-world examples, you can develop the skills and confidence to solve even the most complex exponential equations. Remember to simplify the left-hand side, express both sides with the same base, and equate the exponents. With practice and dedication, you'll become a master of exponential equations in no time!