Simplify The Expression: $y = -9^{-x} + 3$

Understanding the Problem

The given expression is $y = -9^{-x} + 3$. This is an algebraic expression that involves a negative exponent and a constant term. Our goal is to simplify this expression, which means rewriting it in a more manageable form.

The Role of Negative Exponents

Negative exponents are a fundamental concept in algebra. When we have a negative exponent, it means that the base is being raised to the power of the opposite sign. In this case, $9^{-x}$ means that 9 is being raised to the power of $-x$. To simplify this expression, we need to understand the properties of negative exponents.

Properties of Negative Exponents

There are two key properties of negative exponents that we need to remember:

• Property 1: $a^{-n} = \frac{1}{a^n}$

  This property states that a negative exponent can be rewritten as a fraction. In other words, $a^{-n}$ is equal to $\frac{1}{a^n}$.

• Property 2: $a^{-n} = \frac{1}{a^n} = \frac{1}{a} \cdot \frac{1}{a} \cdots \frac{1}{a}$ (n times)

  This property states that a negative exponent can be rewritten as a product of fractions. In other words, $a^{-n}$ is equal to $\frac{1}{a} \cdot \frac{1}{a} \cdots \frac{1}{a}$ (n times).

Simplifying the Expression

Now that we have a good understanding of negative exponents, let's simplify the expression $y = -9^{-x} + 3$.

Using Property 1, we can rewrite $9^{-x}$ as $\frac{1}{9^x}$.

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the expression
expr = -9**(-x) + 3

# Simplify the expression
simplified_expr = sp.simplify(expr)

print(simplified_expr)

This code will output the simplified expression.

The Final Answer

The final answer is $\boxed{-\frac{1}{9^x} + 3}$.

Conclusion

In this article, we simplified the expression $y = -9^{-x} + 3$ using the properties of negative exponents. We learned that a negative exponent can be rewritten as a fraction or a product of fractions. We also saw how to use Python code to simplify the expression. The final answer is $\boxed{-\frac{1}{9^x} + 3}$.

Further Reading

If you want to learn more about negative exponents and algebraic expressions, I recommend checking out the following resources:

• Khan Academy: Algebra
• MIT OpenCourseWare: Algebra
• Wolfram MathWorld: Negative Exponent

Final Thoughts

Simplifying algebraic expressions is an essential skill in mathematics. By understanding the properties of negative exponents, we can rewrite complex expressions in a more manageable form. I hope this article has been helpful in simplifying the expression $y = -9^{-x} + 3$. If you have any questions or comments, please feel free to leave them below.

Understanding the Problem

The given expression is $y = -9^{-x} + 3$. This is an algebraic expression that involves a negative exponent and a constant term. Our goal is to simplify this expression, which means rewriting it in a more manageable form.

Q&A

Q: What is a negative exponent?

A: A negative exponent is a mathematical operation that involves raising a number to the power of a negative number. In the expression $y = -9^{-x} + 3$, the negative exponent is $-x$.

Q: How do I simplify a negative exponent?

A: To simplify a negative exponent, you can use the property $a^{-n} = \frac{1}{a^n}$. This means that $a^{-n}$ is equal to $\frac{1}{a^n}$.

Q: Can you give an example of simplifying a negative exponent?

A: Yes, let's simplify the expression $9^{-x}$. Using the property $a^{-n} = \frac{1}{a^n}$, we can rewrite $9^{-x}$ as $\frac{1}{9^x}$.

Q: How do I simplify the expression $y = -9^{-x} + 3$?

A: To simplify the expression $y = -9^{-x} + 3$, we can use the property $a^{-n} = \frac{1}{a^n}$. This means that $9^{-x}$ is equal to $\frac{1}{9^x}$. Therefore, the simplified expression is $y = -\frac{1}{9^x} + 3$.

Q: Can you use Python code to simplify the expression?

A: Yes, we can use Python code to simplify the expression. Here is an example of how to do it:

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the expression
expr = -9**(-x) + 3

# Simplify the expression
simplified_expr = sp.simplify(expr)

print(simplified_expr)

This code will output the simplified expression.

Q: What is the final answer?

A: The final answer is $\boxed{-\frac{1}{9^x} + 3}$.

Conclusion

In this article, we simplified the expression $y = -9^{-x} + 3$ using the properties of negative exponents. We also answered some common questions about simplifying negative exponents and provided Python code to simplify the expression. The final answer is $\boxed{-\frac{1}{9^x} + 3}$.

Further Reading

If you want to learn more about negative exponents and algebraic expressions, I recommend checking out the following resources:

• Khan Academy: Algebra
• MIT OpenCourseWare: Algebra
• Wolfram MathWorld: Negative Exponent

Final Thoughts

Simplifying algebraic expressions is an essential skill in mathematics. By understanding the properties of negative exponents, we can rewrite complex expressions in a more manageable form. I hope this article has been helpful in simplifying the expression $y = -9^{-x} + 3$. If you have any questions or comments, please feel free to leave them below.