Evaluate $y=\left(\frac{1}{5}\right)^x$ When $x =-3$.A. $y=\frac{1}{125}$ B. $y=-\frac{1}{125}$ C. $y=125$

Introduction

Exponential functions are a fundamental concept in mathematics, and understanding how to evaluate them is crucial for solving various mathematical problems. In this article, we will focus on evaluating the exponential function $y=\left(\frac{1}{5}\right)^x$ when $x =-3$. We will break down the process step by step and provide a clear explanation of the calculations involved.

Understanding Exponential Functions

An exponential function is a mathematical function of the form $y=a^x$, where $a$ is a positive real number and $x$ is the variable. The base $a$ determines the rate at which the function grows or decays. In the case of the function $y=\left(\frac{1}{5}\right)^x$, the base is $\frac{1}{5}$, which is a fraction between 0 and 1.

Evaluating the Function

To evaluate the function $y=\left(\frac{1}{5}\right)^x$ when $x =-3$, we need to substitute $x$ with $-3$ into the function. This gives us:

$$y=\left(\frac{1}{5}\right)^{-3}$$

Using the Properties of Exponents

When dealing with negative exponents, we can use the property that states $a^{-n}=\frac{1}{a^n}$. Applying this property to our function, we get:

$$y=\frac{1}{\left(\frac{1}{5}\right)^3}$$

Simplifying the Expression

To simplify the expression, we need to evaluate the denominator. Using the property of exponents that states $\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$, we can rewrite the denominator as:

$$\left(\frac{1}{5}\right)^3=\frac{1^3}{5^3}=\frac{1}{125}$$

Substituting the Simplified Expression

Now that we have simplified the denominator, we can substitute it back into the original expression:

$$y=\frac{1}{\frac{1}{125}}$$

Using the Property of Inverse Operations

When dealing with fractions, we can use the property that states $\frac{1}{\frac{a}{b}}=\frac{b}{a}$. Applying this property to our expression, we get:

$$y=\frac{125}{1}=125$$

Conclusion

In conclusion, evaluating the exponential function $y=\left(\frac{1}{5}\right)^x$ when $x =-3$ involves using the properties of exponents and simplifying the expression. By following the steps outlined in this article, we have shown that the correct answer is $y=125$.

Common Mistakes to Avoid

When evaluating exponential functions, it's essential to avoid common mistakes such as:

• Not using the correct properties of exponents
• Not simplifying the expression correctly
• Not using the correct inverse operations

By being aware of these common mistakes, you can ensure that you evaluate exponential functions accurately and confidently.

Real-World Applications

Exponential functions have numerous real-world applications, including:

• Modeling population growth and decay
• Describing chemical reactions
• Analyzing financial data

By understanding how to evaluate exponential functions, you can apply this knowledge to solve real-world problems and make informed decisions.

Final Thoughts

Introduction

In our previous article, we discussed how to evaluate the exponential function $y=\left(\frac{1}{5}\right)^x$ when $x =-3$. We also covered the properties of exponents and how to simplify expressions. In this article, we will provide a Q&A guide to help you better understand how to evaluate exponential functions.

Q: What is an exponential function?

A: An exponential function is a mathematical function of the form $y=a^x$, where $a$ is a positive real number and $x$ is the variable.

Q: What is the base of an exponential function?

A: The base of an exponential function is the number $a$ that is raised to the power of $x$. In the case of the function $y=\left(\frac{1}{5}\right)^x$, the base is $\frac{1}{5}$.

Q: How do I evaluate an exponential function with a negative exponent?

A: To evaluate an exponential function with a negative exponent, you can use the property that states $a^{-n}=\frac{1}{a^n}$. For example, to evaluate the function $y=\left(\frac{1}{5}\right)^{-3}$, you can rewrite it as $y=\frac{1}{\left(\frac{1}{5}\right)^3}$.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, you can use the property that states $\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$. For example, to simplify the expression $\left(\frac{1}{5}\right)^3$, you can rewrite it as $\frac{1^3}{5^3}=\frac{1}{125}$.

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

A: An exponential function is a function of the form $y=a^x$, where $a$ is a positive real number and $x$ is the variable. A power function, on the other hand, is a function of the form $y=ax^n$, where $a$ and $n$ are constants.

Q: How do I use the properties of exponents to simplify an expression?

A: To simplify an expression using the properties of exponents, you can use the following rules:

• $a^m \cdot a^n = a^{m+n}$
• $\frac{a^m}{a^n} = a^{m-n}$
• $(a^m)^n = a^{mn}$

Q: What are some common mistakes to avoid when evaluating exponential functions?

A: Some common mistakes to avoid when evaluating exponential functions include:

• Not using the correct properties of exponents
• Not simplifying the expression correctly
• Not using the correct inverse operations

Q: How do I apply exponential functions to real-world problems?

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

• Modeling population growth and decay
• Describing chemical reactions
• Analyzing financial data

By understanding how to evaluate exponential functions, you can apply this knowledge to solve real-world problems and make informed decisions.

Conclusion

In conclusion, evaluating exponential functions is a critical skill in mathematics, and understanding how to do it correctly is essential for solving various mathematical problems. By following the steps outlined in this article and using the properties of exponents, you can evaluate exponential functions with confidence and accuracy. Remember to avoid common mistakes and apply exponential functions to real-world problems to make informed decisions.