Complete The Tables Of Values.${ \begin{array}{|c|c|} \hline x & 4^{-x} \ \hline -1 & 4 \ \hline 0 & A \ \hline 2 & B \ \hline 4 & C \ \hline \end{array} }$[ \begin{array}{|c|c|} \hline x & \left(\frac{2}{3}\right)^x \ \hline -1

Mar 8, 2025 by ADMIN 231 views

**Completing Tables of Values: A Mathematical Exploration**

Introduction

In mathematics, tables of values are a fundamental concept used to represent the relationship between two variables. These tables are essential in various mathematical operations, including algebra, geometry, and calculus. In this article, we will focus on completing two tables of values, one for the expression $4^{-x}$ and the other for the expression $\left(\frac{2}{3}\right)^x$ . We will explore the mathematical concepts behind these expressions and provide step-by-step solutions to complete the tables.

Table 1: Completing the Values of $4^{-x}$

The first table represents the values of the expression $4^{-x}$ for different values of $x$ . We are given the values of $x$ as $-1$ , $0$ , $2$ , and $4$ , and we need to find the corresponding values of $4^{-x}$ .

$x$	$4^{-x}$
-1	4
0	a
2	b
4	c

To complete the table, we need to understand the concept of negative exponents. When we have a negative exponent, it means that we are taking the reciprocal of the base raised to the power of the positive exponent. In this case, $4^{-x}$ is equivalent to $\frac{1}{4^x}$ .

Let's start by finding the value of $4^{-1}$ .

4^{-1} = \frac{1}{4^1} = \frac{1}{4}

Now, we can find the value of $4^0$ .

4^0 = 1

Next, we need to find the value of $4^2$ .

4^2 = 16

Finally, we need to find the value of $4^4$ .

4^4 = 256

Now that we have the values of $4^x$ for different values of $x$ , we can find the corresponding values of $4^{-x}$ by taking the reciprocal.

$x$	$4^{-x}$
-1	4
0	1
2	$\frac{1}{16}$
4	$\frac{1}{256}$

Therefore, the completed table of values for the expression $4^{-x}$ is:

$x$	$4^{-x}$
-1	4
0	1
2	$\frac{1}{16}$
4	$\frac{1}{256}$

Table 2: Completing the Values of $\left(\frac{2}{3}\right)^x$

The second table represents the values of the expression $\left(\frac{2}{3}\right)^x$ for different values of $x$ . We are given the value of $x$ as $-1$ , and we need to find the corresponding value of $\left(\frac{2}{3}\right)^x$ .

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

To complete the table, we need to understand the concept of exponentiation. When we have a variable in the exponent, it means that we are raising the base to the power of the variable.

Let's start by finding the value of $\left(\frac{2}{3}\right)^{-1}$ .

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

Therefore, the completed table of values for the expression $\left(\frac{2}{3}\right)^x$ is:

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

Conclusion

In this article, we completed two tables of values, one for the expression $4^{-x}$ and the other for the expression $\left(\frac{2}{3}\right)^x$ . We used the concepts of negative exponents and exponentiation to find the corresponding values of the expressions. The completed tables of values are essential in various mathematical operations, including algebra, geometry, and calculus.

References

[1] Khan Academy. (n.d.). Exponents and Exponent Rules. Retrieved from https://www.khanacademy.org/math/algebra/x2f6b7d7d-exponents-and-exponent-rules
[2] Math Open Reference. (n.d.). Exponents and Powers. Retrieved from https://www.mathopenref.com/exponents.html

Introduction

In our previous article, we explored the concept of completing tables of values for the expressions $4^{-x}$ and $\left(\frac{2}{3}\right)^x$ . We provided step-by-step solutions to complete the tables and discussed the mathematical concepts behind these expressions. In this article, we will answer some frequently asked questions related to completing tables of values.

Q: What is the difference between a table of values and a graph?

A: A table of values is a list of values of a function for specific input values, while a graph is a visual representation of the function. A table of values provides a more detailed and precise representation of the function, whereas a graph provides a visual representation of the function's behavior.

Q: How do I determine the values of a table of values?

A: To determine the values of a table of values, you need to understand the mathematical concept behind the expression. For example, if the expression is $4^{-x}$ , you need to understand the concept of negative exponents and how to take the reciprocal of the base raised to the power of the positive exponent.

Q: What is the significance of completing tables of values?

A: Completing tables of values is essential in various mathematical operations, including algebra, geometry, and calculus. It helps to understand the behavior of functions and how they change as the input values change. It also helps to identify patterns and relationships between variables.

Q: How do I use tables of values to solve problems?

A: To use tables of values to solve problems, you need to understand the mathematical concept behind the expression and how to apply it to the problem. For example, if you are given a table of values for the expression $4^{-x}$ and you need to find the value of $4^{-2}$ , you can use the table to find the value.

Q: Can I use tables of values to graph functions?

A: Yes, you can use tables of values to graph functions. By plotting the points in the table, you can create a graph of the function. This is a useful technique for visualizing the behavior of functions and identifying patterns and relationships between variables.

Q: What are some common mistakes to avoid when completing tables of values?

A: Some common mistakes to avoid when completing tables of values include:

Not understanding the mathematical concept behind the expression
Not following the correct order of operations
Not using the correct notation and terminology
Not checking the work for errors

Q: How can I practice completing tables of values?

A: You can practice completing tables of values by working through examples and exercises in your textbook or online resources. You can also try creating your own tables of values for different expressions and functions.

Conclusion

In this article, we answered some frequently asked questions related to completing tables of values. We discussed the significance of completing tables of values, how to determine the values of a table of values, and how to use tables of values to solve problems. We also provided some common mistakes to avoid and tips for practicing completing tables of values.

References

[1] Khan Academy. (n.d.). Exponents and Exponent Rules. Retrieved from https://www.khanacademy.org/math/algebra/x2f6b7d7d-exponents-and-exponent-rules
[2] Math Open Reference. (n.d.). Exponents and Powers. Retrieved from https://www.mathopenref.com/exponents.html

Complete The Tables Of Values.${ \begin{array}{|c|c|} \hline x & 4^{-x} \ \hline -1 & 4 \ \hline 0 & A \ \hline 2 & B \ \hline 4 & C \ \hline \end{array} }$[ \begin{array}{|c|c|} \hline x & \left(\frac{2}{3}\right)^x \ \hline -1

Introduction

Table 1: Completing the Values of $4^{-x}$

Table 2: Completing the Values of $\left(\frac{2}{3}\right)^x$

Conclusion

References

Further Reading

Introduction

Q: What is the difference between a table of values and a graph?

Q: How do I determine the values of a table of values?

Q: What is the significance of completing tables of values?

Q: How do I use tables of values to solve problems?

Q: Can I use tables of values to graph functions?

Q: What are some common mistakes to avoid when completing tables of values?

Q: How can I practice completing tables of values?

Conclusion

References

Further Reading

Introduction

Table 1: Completing the Values of 4−x4^{-x}4−x

Table 2: Completing the Values of (23)x\left(\frac{2}{3}\right)^x(32​)x

Conclusion

References

Further Reading

Introduction

Q: What is the difference between a table of values and a graph?

Q: How do I determine the values of a table of values?

Q: What is the significance of completing tables of values?

Q: How do I use tables of values to solve problems?

Q: Can I use tables of values to graph functions?

Q: What are some common mistakes to avoid when completing tables of values?

Q: How can I practice completing tables of values?

Conclusion

References

Further Reading

Table 1: Completing the Values of $4^{-x}$

Table 2: Completing the Values of $\left(\frac{2}{3}\right)^x$