Use The Like-bases Property And Exponents To Solve The Equation:$\left(\frac{1}{5}\right)^{n-9}=5^{5n-10}$Solve For $n$.

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of exponents. In this article, we will explore how to use the like-bases property and exponents to solve the equation $\left(\frac{1}{5}\right)^{n-9}=5^{5n-10}$. We will break down the solution step by step, providing a clear and concise explanation of each step.

Understanding the Like-Bases Property

The like-bases property states that if two exponential expressions have the same base, then their exponents can be set equal to each other. In other words, if $a^x = a^y$, then $x = y$. This property is a powerful tool for solving exponential equations.

Rewriting the Equation

To solve the equation $\left(\frac{1}{5}\right)^{n-9}=5^{5n-10}$, we need to rewrite it in a way that allows us to use the like-bases property. We can do this by expressing $\frac{1}{5}$ as a power of 5.

$\left(\frac{1}{5}\right)^{n-9} = \left(5^{-1}\right)^{n-9}$

Now, we can use the power of a power property, which states that $\left(a^m\right)^n = a^{mn}$. Applying this property, we get:

$\left(5^{-1}\right)^{n-9} = 5^{(-1)(n-9)}$

Simplifying the exponent, we get:

$5^{(-1)(n-9)} = 5^{-n+9}$

Applying the Like-Bases Property

Now that we have rewritten the equation, we can apply the like-bases property. We set the exponents equal to each other:

$-n+9 = 5n-10$

Solving for $n$

To solve for $n$, we need to isolate the variable on one side of the equation. We can do this by adding $n$ to both sides of the equation:

$9 = 6n-10$

Next, we add 10 to both sides of the equation:

$19 = 6n$

Finally, we divide both sides of the equation by 6:

$n = \frac{19}{6}$

Conclusion

In this article, we used the like-bases property and exponents to solve the equation $\left(\frac{1}{5}\right)^{n-9}=5^{5n-10}$. We rewrote the equation in a way that allowed us to use the like-bases property, and then applied the property to solve for $n$. The solution was $n = \frac{19}{6}$.

Tips and Tricks

• When solving exponential equations, it's essential to use the like-bases property to rewrite the equation in a way that allows you to set the exponents equal to each other.
• Make sure to apply the power of a power property when rewriting the equation.
• When solving for the variable, isolate it on one side of the equation by adding or subtracting the same value from both sides.

Common Mistakes

• Failing to rewrite the equation in a way that allows you to use the like-bases property.
• Not applying the power of a power property when rewriting the equation.
• Not isolating the variable on one side of the equation when solving for it.

Real-World Applications

Exponential equations have numerous real-world applications, including:

• Modeling population growth and decline
• Calculating compound interest
• Determining the half-life of radioactive substances
• Solving problems in physics and engineering

Conclusion

Introduction

In our previous article, we explored how to use the like-bases property and exponents to solve the equation $\left(\frac{1}{5}\right)^{n-9}=5^{5n-10}$. We broke down the solution step by step, providing a clear and concise explanation of each step. In this article, we will answer some of the most frequently asked questions about solving exponential equations with the like-bases property.

Q: What is the like-bases property?

A: The like-bases property states that if two exponential expressions have the same base, then their exponents can be set equal to each other. In other words, if $a^x = a^y$, then $x = y$.

Q: How do I apply the like-bases property to solve an exponential equation?

A: To apply the like-bases property, you need to rewrite the equation in a way that allows you to set the exponents equal to each other. This may involve expressing the base as a power of a different base, or using the power of a power property to simplify the equation.

Q: What is the power of a power property?

A: The power of a power property states that $\left(a^m\right)^n = a^{mn}$. This means that when you raise a power to a power, you can multiply the exponents.

Q: How do I solve for the variable in an exponential equation?

A: To solve for the variable, you need to isolate it on one side of the equation. This may involve adding or subtracting the same value from both sides, or multiplying or dividing both sides by the same value.

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 rewrite the equation in a way that allows you to use the like-bases property
• Not applying the power of a power property when rewriting the equation
• Not isolating the variable on one side of the equation when solving for it

Q: What are some real-world applications of exponential equations?

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

• Modeling population growth and decline
• Calculating compound interest
• Determining the half-life of radioactive substances
• Solving problems in physics and engineering

Q: Can you provide an example of how to solve an exponential equation using the like-bases property?

A: Let's consider the equation $\left(\frac{1}{2}\right)^{x-3}=2^{3x-2}$. To solve this equation, we can rewrite it as follows:

$\left(\frac{1}{2}\right)^{x-3} = \left(2^{-1}\right)^{x-3}$

Using the power of a power property, we can simplify the equation as follows:

$2^{-x+3} = 2^{3x-2}$

Now, we can set the exponents equal to each other:

$-x+3 = 3x-2$

Solving for $x$, we get:

$x = \frac{5}{4}$

Conclusion

In this article, we answered some of the most frequently asked questions about solving exponential equations with the like-bases property. We provided examples and explanations to help you understand the concepts and apply them to real-world problems. By following the steps outlined in this article, you can solve complex exponential equations and apply the concepts to a wide range of applications.

Tips and Tricks

• Make sure to rewrite the equation in a way that allows you to use the like-bases property.
• Apply the power of a power property when rewriting the equation.
• Isolate the variable on one side of the equation when solving for it.
• Use real-world examples to help you understand the concepts and apply them to different situations.

Common Mistakes

• Failing to rewrite the equation in a way that allows you to use the like-bases property.
• Not applying the power of a power property when rewriting the equation.
• Not isolating the variable on one side of the equation when solving for it.

Real-World Applications

Exponential equations have numerous real-world applications, including:

• Modeling population growth and decline
• Calculating compound interest
• Determining the half-life of radioactive substances
• Solving problems in physics and engineering

Conclusion

In conclusion, solving exponential equations with the like-bases property and exponents requires a deep understanding of the properties of exponents. By following the steps outlined in this article, you can solve complex exponential equations and apply the concepts to a wide range of applications.