Solve The Equation:${ 81^{3x-6} = 729^{x+4} }$

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Introduction

In this article, we will delve into solving the equation $81^{3x-6} = 729^{x+4}$. This equation involves exponentiation and can be solved using various mathematical techniques. We will break down the solution step by step, making it easy to understand and follow.

Understanding the Equation

The given equation is $81^{3x-6} = 729^{x+4}$. To solve this equation, we need to first understand the properties of exponents and how to manipulate them. The base of the exponent is 81 and 729, which can be expressed as powers of 3: $81 = 3^4$ and $729 = 3^6$.

Simplifying the Equation

Using the properties of exponents, we can rewrite the equation as $(3^4)^{3x-6} = (3^6)^{x+4}$. This simplifies to $3^{12x-24} = 3^{6x+24}$.

Setting Up an Equation

Since the bases are the same, we can set up an equation by equating the exponents: $12x-24 = 6x+24$.

Solving for x

To solve for x, we need to isolate the variable. We can do this by subtracting 6x from both sides of the equation: $6x-24 = 24$. Then, we add 24 to both sides: $6x = 48$. Finally, we divide both sides by 6: $x = 8$.

Verifying the Solution

To verify the solution, we can substitute x = 8 back into the original equation: $81^{3(8)-6} = 729^{8+4}$. This simplifies to $81^{18} = 729^{12}$. Since $81 = 3^4$ and $729 = 3^6$, we can rewrite the equation as $(3^4)^{18} = (3^6)^{12}$. This further simplifies to $3^{72} = 3^{72}$, which is true.

Conclusion

In this article, we solved the equation $81^{3x-6} = 729^{x+4}$ using the properties of exponents and algebraic manipulation. We simplified the equation, set up an equation by equating the exponents, and solved for x. We then verified the solution by substituting x = 8 back into the original equation. The final answer is x = 8.

Additional Tips and Tricks

• When solving equations involving exponents, it's essential to understand the properties of exponents and how to manipulate them.
• Use algebraic manipulation to simplify the equation and set up an equation by equating the exponents.
• Verify the solution by substituting the value back into the original equation.

Frequently Asked Questions

• Q: What is the base of the exponent in the given equation? A: The base of the exponent is 81 and 729, which can be expressed as powers of 3: $81 = 3^4$ and $729 = 3^6$.
• Q: How do we simplify the equation? A: We can simplify the equation by using the properties of exponents and rewriting the equation as $(3^4)^{3x-6} = (3^6)^{x+4}$.
• Q: How do we solve for x? A: We can solve for x by setting up an equation by equating the exponents and then isolating the variable.

Related Topics

• Exponents and their properties
• Algebraic manipulation and equation solving
• Mathematical techniques for solving equations involving exponents

References

• [1] Khan Academy. (n.d.). Exponents and Exponent Rules. Retrieved from https://www.khanacademy.org/math/algebra/x2f6f7d/x2f6f8d
• [2] Mathway. (n.d.). Algebra Solver. Retrieved from https://www.mathway.com/

Final Thoughts

Solving the equation $81^{3x-6} = 729^{x+4}$ requires a deep understanding of the properties of exponents and algebraic manipulation. By following the steps outlined in this article, you can solve this equation and gain a better understanding of mathematical techniques for solving equations involving exponents.

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Introduction

In our previous article, we solved the equation $81^{3x-6} = 729^{x+4}$ using the properties of exponents and algebraic manipulation. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on solving equations involving exponents.

Q&A

Q: What is the base of the exponent in the given equation?

A: The base of the exponent is 81 and 729, which can be expressed as powers of 3: $81 = 3^4$ and $729 = 3^6$.

Q: How do we simplify the equation?

A: We can simplify the equation by using the properties of exponents and rewriting the equation as $(3^4)^{3x-6} = (3^6)^{x+4}$. This simplifies to $3^{12x-24} = 3^{6x+24}$.

Q: How do we solve for x?

A: We can solve for x by setting up an equation by equating the exponents and then isolating the variable. In this case, we have $12x-24 = 6x+24$. We can solve for x by subtracting 6x from both sides: $6x-24 = 24$. Then, we add 24 to both sides: $6x = 48$. Finally, we divide both sides by 6: $x = 8$.

Q: What if the equation has a different base?

A: If the equation has a different base, we can still use the properties of exponents to simplify the equation. For example, if the equation is $2^{3x-6} = 8^{x+4}$, we can rewrite the equation as $(2^3)^{3x-6} = (2^3)^{x+4}$. This simplifies to $2^{9x-18} = 2^{3x+12}$.

Q: Can we use logarithms to solve the equation?

A: Yes, we can use logarithms to solve the equation. For example, if the equation is $2^{3x-6} = 8^{x+4}$, we can take the logarithm of both sides: $\log(2^{3x-6}) = \log(8^{x+4})$. This simplifies to $(3x-6)\log(2) = (x+4)\log(8)$.

Q: What if the equation has a negative exponent?

A: If the equation has a negative exponent, we can still use the properties of exponents to simplify the equation. For example, if the equation is $2^{-3x+6} = 8^{-x-4}$, we can rewrite the equation as $(2^{-3})^{-3x+6} = (2^{-3})^{-x-4}$. This simplifies to $2^{9x-18} = 2^{3x+12}$.

Q: Can we use a calculator to solve the equation?

A: Yes, we can use a calculator to solve the equation. For example, if the equation is $2^{3x-6} = 8^{x+4}$, we can use a calculator to find the value of x.

Conclusion

In this article, we provided a Q&A section to help clarify any doubts and provide additional information on solving equations involving exponents. We covered topics such as simplifying the equation, solving for x, and using logarithms and calculators to solve the equation.

Additional Tips and Tricks

• When solving equations involving exponents, it's essential to understand the properties of exponents and how to manipulate them.
• Use algebraic manipulation to simplify the equation and set up an equation by equating the exponents.
• Verify the solution by substituting the value back into the original equation.
• Use logarithms and calculators to solve the equation.

Frequently Asked Questions

• Q: What is the base of the exponent in the given equation? A: The base of the exponent is 81 and 729, which can be expressed as powers of 3: $81 = 3^4$ and $729 = 3^6$.
• Q: How do we simplify the equation? A: We can simplify the equation by using the properties of exponents and rewriting the equation as $(3^4)^{3x-6} = (3^6)^{x+4}$.
• Q: How do we solve for x? A: We can solve for x by setting up an equation by equating the exponents and then isolating the variable.

Related Topics

• Exponents and their properties
• Algebraic manipulation and equation solving
• Mathematical techniques for solving equations involving exponents
• Logarithms and their properties
• Calculators and their use in solving equations

References

• [1] Khan Academy. (n.d.). Exponents and Exponent Rules. Retrieved from https://www.khanacademy.org/math/algebra/x2f6f7d/x2f6f8d
• [2] Mathway. (n.d.). Algebra Solver. Retrieved from https://www.mathway.com/

Final Thoughts

Solving equations involving exponents requires a deep understanding of the properties of exponents and algebraic manipulation. By following the steps outlined in this article and using the Q&A section, you can solve equations involving exponents and gain a better understanding of mathematical techniques for solving equations.