Solve For { X $} : : : { \left(\frac{1}{8}\right)^{4x-9} = 64^{3x+1} \}

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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, namely the equation (18)4x−9=643x+1\left(\frac{1}{8}\right)^{4x-9} = 64^{3x+1}. This equation involves two exponential expressions with different bases, and our goal is to find the value of xx that satisfies this equation.

Understanding Exponential Equations

Before we dive into the solution, let's take a moment to understand the basics of exponential equations. An exponential equation is an equation that involves an exponential expression, which is a number raised to a power. In this case, we have two exponential expressions: (18)4x−9\left(\frac{1}{8}\right)^{4x-9} and 643x+164^{3x+1}. The base of an exponential expression is the number that is being raised to a power, and the exponent is the power to which the base is being raised.

Simplifying the Equation

To solve this equation, we need to simplify it first. We can start by rewriting the equation in a more manageable form. We know that 64=8264 = 8^2, so we can rewrite the equation as (18)4x−9=(82)3x+1\left(\frac{1}{8}\right)^{4x-9} = (8^2)^{3x+1}.

Using Exponent Rules to Simplify

Now, let's use some exponent rules to simplify the equation. We know that (am)n=amn(a^m)^n = a^{mn}, so we can rewrite the equation as (18)4x−9=82(3x+1)\left(\frac{1}{8}\right)^{4x-9} = 8^{2(3x+1)}.

Simplifying Further

We can simplify the equation further by using the rule (1a)n=a−n\left(\frac{1}{a}\right)^n = a^{-n}. Applying this rule to the left-hand side of the equation, we get 8−(4x−9)=82(3x+1)8^{-(4x-9)} = 8^{2(3x+1)}.

Equating Exponents

Now that we have simplified the equation, we can equate the exponents on both sides. This gives us the equation −(4x−9)=2(3x+1)-(4x-9) = 2(3x+1).

Solving for x

To solve for xx, we need to isolate the variable on one side of the equation. We can start by distributing the 2 on the right-hand side of the equation, which gives us −(4x−9)=6x+2-(4x-9) = 6x+2.

Simplifying the Equation

Next, we can simplify the equation by combining like terms. We can add 4x4x to both sides of the equation, which gives us −9=10x+2-9 = 10x+2.

Isolating x

Now that we have simplified the equation, we can isolate xx by subtracting 2 from both sides of the equation. This gives us −11=10x-11 = 10x.

Solving for x

Finally, we can solve for xx by dividing both sides of the equation by 10. This gives us x=−1110x = -\frac{11}{10}.

Conclusion

In this article, we have solved a specific type of exponential equation, namely the equation (18)4x−9=643x+1\left(\frac{1}{8}\right)^{4x-9} = 64^{3x+1}. We have used various exponent rules and simplification techniques to isolate the variable xx and find its value. The final answer is x=−1110x = -\frac{11}{10}.

Additional Tips and Tricks

  • When solving exponential equations, it's essential to simplify the equation first by using exponent rules and simplification techniques.
  • Make sure to equate the exponents on both sides of the equation and isolate the variable on one side.
  • Use algebraic manipulations to solve for the variable, and don't be afraid to use inverse operations to isolate the variable.

Frequently Asked Questions

  • Q: What is the main difference between exponential equations and linear equations? A: Exponential equations involve exponential expressions, while linear equations involve linear expressions.
  • Q: How do I simplify an exponential equation? A: You can simplify an exponential equation by using exponent rules and simplification techniques, such as rewriting the equation in a more manageable form or using the rule (1a)n=a−n\left(\frac{1}{a}\right)^n = a^{-n}.
  • Q: What is the final answer to the equation (18)4x−9=643x+1\left(\frac{1}{8}\right)^{4x-9} = 64^{3x+1}? A: The final answer is x=−1110x = -\frac{11}{10}.

References

Introduction

In our previous article, we solved a specific type of exponential equation, namely the equation (18)4x−9=643x+1\left(\frac{1}{8}\right)^{4x-9} = 64^{3x+1}. We used various exponent rules and simplification techniques to isolate the variable xx and find its value. In this article, we will provide a Q&A guide to help you better understand exponential equations and how to solve them.

Q&A: Exponential Equations

Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential expression, which is a number raised to a power.

Q: How do I simplify an exponential equation?

A: You can simplify an exponential equation by using exponent rules and simplification techniques, such as rewriting the equation in a more manageable form or using the rule (1a)n=a−n\left(\frac{1}{a}\right)^n = a^{-n}.

Q: What is the main difference between exponential equations and linear equations?

A: Exponential equations involve exponential expressions, while linear equations involve linear expressions.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the variable on one side of the equation. You can do this by using algebraic manipulations, such as adding or subtracting the same value to both sides of the equation, or multiplying or dividing both sides of the equation by the same value.

Q: What is the final answer to the equation (18)4x−9=643x+1\left(\frac{1}{8}\right)^{4x-9} = 64^{3x+1}?

A: The final answer is x=−1110x = -\frac{11}{10}.

Q: Can I use logarithms to solve exponential equations?

A: Yes, you can use logarithms to solve exponential equations. Logarithms can help you isolate the variable on one side of the equation by converting the exponential expression into a linear expression.

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

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

  • Not simplifying the equation before solving it
  • Not isolating the variable on one side of the equation
  • Not using the correct exponent rules
  • Not checking the solution to make sure it satisfies the original equation

Q: How do I check my solution to an exponential equation?

A: To check your solution to an exponential equation, you need to plug the value of the variable back into the original equation and make sure it is true. If the equation is true, then your solution is correct.

Additional Tips and Tricks

  • When solving exponential equations, it's essential to simplify the equation first by using exponent rules and simplification techniques.
  • Make sure to equate the exponents on both sides of the equation and isolate the variable on one side.
  • Use algebraic manipulations to solve for the variable, and don't be afraid to use inverse operations to isolate the variable.
  • Check your solution to make sure it satisfies the original equation.

Frequently Asked Questions

  • Q: What is the difference between an exponential equation and a logarithmic equation? A: An exponential equation involves an exponential expression, while a logarithmic equation involves a logarithmic expression.
  • Q: How do I solve a logarithmic equation? A: To solve a logarithmic equation, you need to isolate the variable on one side of the equation. You can do this by using algebraic manipulations, such as adding or subtracting the same value to both sides of the equation, or multiplying or dividing both sides of the equation by the same value.
  • Q: Can I use exponential equations to model real-world problems? A: Yes, you can use exponential equations to model real-world problems. Exponential equations can be used to model population growth, chemical reactions, and other phenomena that involve exponential change.

References