What Value Of $n$ Makes The Equation True?$\frac{(-5)^{11}}{(-5)^n}=(-5)^2 \cdot(-5)^1$Show Your Work.

Introduction

In this article, we will explore the concept of exponents and how to solve equations involving them. We will examine the given equation $\frac{(-5)^{11}}{(-5)^n}=(-5)^2 \cdot(-5)^1$ and determine the value of $n$ that makes the equation true.

Understanding Exponents

Exponents are a shorthand way of writing repeated multiplication. For example, $a^b$ means $a$ multiplied by itself $b$ times. In the given equation, we have $(-5)^{11}$, which means $-5$ multiplied by itself $11$ times.

Simplifying the Equation

To simplify the equation, we can start by evaluating the right-hand side. We have $(-5)^2 \cdot(-5)^1$, which can be rewritten as $(-5)^{2+1}$. Using the rule of exponents that states $a^b \cdot a^c = a^{b+c}$, we can simplify this to $(-5)^3$.

import math

# Define the variables
a = -5
b = 2
c = 1

# Calculate the right-hand side
rhs = a ** (b + c)
print(rhs)

Evaluating the Left-Hand Side

Now, let's evaluate the left-hand side of the equation. We have $\frac{(-5)^{11}}{(-5)^n}$. Using the rule of exponents that states $\frac{a^b}{a^c} = a^{b-c}$, we can simplify this to $(-5)^{11-n}$.

# Define the variables
a = -5
b = 11
c = n

# Calculate the left-hand side
lhs = a ** (b - c)
print(lhs)

Equating the Two Sides

Now that we have simplified both sides of the equation, we can equate them. We have $(-5)^{11-n} = (-5)^3$. Since the bases are the same, we can equate the exponents. This gives us $11-n = 3$.

# Define the variables
a = 11
b = n
c = 3

# Equate the exponents
equation = a - b == c
print(equation)

Solving for n

Now that we have the equation $11-n = 3$, we can solve for $n$. Subtracting $11$ from both sides gives us $-n = -8$. Multiplying both sides by $-1$ gives us $n = 8$.

# Define the variables
a = 11
b = 3

# Solve for n
n = a - b
print(n)

Conclusion

In this article, we have explored the concept of exponents and how to solve equations involving them. We have examined the given equation $\frac{(-5)^{11}}{(-5)^n}=(-5)^2 \cdot(-5)^1$ and determined the value of $n$ that makes the equation true. We have shown that $n = 8$.

Final Answer

The final answer is $\boxed{8}$.

Additional Resources

For more information on exponents and how to solve equations involving them, please see the following resources:

• Khan Academy: Exponents and Exponential Functions
• Mathway: Exponents and Exponential Functions
• Wolfram Alpha: Exponents and Exponential Functions

References

• "Algebra and Trigonometry" by Michael Sullivan
• "College Algebra" by James Stewart
• "Mathematics for the Nonmathematician" by Morris Kline
  Q&A: Exponents and Exponential Functions =============================================

Introduction

In our previous article, we explored the concept of exponents and how to solve equations involving them. We examined the equation $\frac{(-5)^{11}}{(-5)^n}=(-5)^2 \cdot(-5)^1$ and determined the value of $n$ that makes the equation true. In this article, we will answer some frequently asked questions about exponents and exponential functions.

Q: What is an exponent?

A: An exponent is a shorthand way of writing repeated multiplication. For example, $a^b$ means $a$ multiplied by itself $b$ times.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you can use the rules of exponents. For example, if you have $a^b \cdot a^c$, you can simplify it to $a^{b+c}$.

Q: What is the rule for dividing exponents?

A: The rule for dividing exponents is $\frac{a^b}{a^c} = a^{b-c}$.

Q: How do I solve an equation with exponents?

A: To solve an equation with exponents, you can use the rules of exponents to simplify the equation and then solve for the variable.

Q: What is the difference between an exponential function and a polynomial function?

A: An exponential function is a function of the form $f(x) = a^x$, where $a$ is a constant. A polynomial function is a function of the form $f(x) = ax^b + cx^d + \ldots$, where $a$, $b$, $c$, $d$, $\ldots$ are constants.

Q: How do I graph an exponential function?

A: To graph an exponential function, you can use a graphing calculator or a computer program. You can also use a table of values to plot the function.

Q: What is the domain and range of an exponential function?

A: The domain of an exponential function is all real numbers, and the range is all positive real numbers.

Q: How do I find the inverse of an exponential function?

A: To find the inverse of an exponential function, you can swap the $x$ and $y$ values and solve for $y$.

Q: What is the derivative of an exponential function?

A: The derivative of an exponential function is the function itself.

Q: How do I integrate an exponential function?

A: To integrate an exponential function, you can use the formula $\int a^x dx = \frac{a^x}{\ln a} + C$.

Conclusion

In this article, we have answered some frequently asked questions about exponents and exponential functions. We hope that this article has been helpful in clarifying any confusion you may have had about these topics.

Additional Resources

For more information on exponents and exponential functions, please see the following resources:

• Khan Academy: Exponents and Exponential Functions
• Mathway: Exponents and Exponential Functions
• Wolfram Alpha: Exponents and Exponential Functions

References

• "Algebra and Trigonometry" by Michael Sullivan
• "College Algebra" by James Stewart
• "Mathematics for the Nonmathematician" by Morris Kline

Q&A Session

Do you have any questions about exponents and exponential functions? Please feel free to ask us, and we will do our best to answer your question.

Q: What is the value of $n$ in the equation $\frac{(-5)^{11}}{(-5)^n}=(-5)^2 \cdot(-5)^1$?

A: The value of $n$ is 8.

Q: How do I simplify the expression $a^b \cdot a^c$?

A: You can simplify the expression $a^b \cdot a^c$ to $a^{b+c}$.

Q: What is the rule for dividing exponents?

A: The rule for dividing exponents is $\frac{a^b}{a^c} = a^{b-c}$.

Q: How do I solve an equation with exponents?

A: To solve an equation with exponents, you can use the rules of exponents to simplify the equation and then solve for the variable.

Q: What is the difference between an exponential function and a polynomial function?

A: An exponential function is a function of the form $f(x) = a^x$, where $a$ is a constant. A polynomial function is a function of the form $f(x) = ax^b + cx^d + \ldots$, where $a$, $b$, $c$, $d$, $\ldots$ are constants.