Jana Is Ordering A List Of Numbers From Least To Greatest: 7 2 , 7 ⋅ 4 5 , 0.8 , ( 5 ) 2 \sqrt{7^2}, \sqrt{7} \cdot \frac{4}{5}, \sqrt{0.8}, (\sqrt{5})^2 7 2 ​ , 7 ​ ⋅ 5 4 ​ , 0.8 ​ , ( 5 ​ ) 2 Which Statement Can Be Used To Create Her List?A. 0.8 \sqrt{0.8} 0.8 ​ Is Less Than 4 5 \frac{4}{5} 5 4 ​ Because

Understanding the Problem

Jana is tasked with ordering a list of numbers from least to greatest. The list consists of four numbers: $\sqrt{7^2}, \sqrt{7} \cdot \frac{4}{5}, \sqrt{0.8}, (\sqrt{5})^2$. To create her list, Jana needs to determine the correct order of these numbers. In this article, we will explore the mathematical concepts and techniques required to order these numbers.

Analyzing the Numbers

Let's start by analyzing each number in the list:

• $\sqrt{7^2}$: This expression represents the square root of $7^2$, which is equal to $7$.
• $\sqrt{7} \cdot \frac{4}{5}$: This expression represents the product of the square root of $7$ and $\frac{4}{5}$.
• $\sqrt{0.8}$: This expression represents the square root of $0.8$.
• $(\sqrt{5})^2$: This expression represents the square of the square root of $5$, which is equal to $5$.

Comparing the Numbers

To determine the correct order of these numbers, we need to compare them. We can start by comparing the first two numbers:

• $\sqrt{7^2}$ is equal to $7$.
• $\sqrt{7} \cdot \frac{4}{5}$ is less than $7$ because $\sqrt{7}$ is less than $7$ and $\frac{4}{5}$ is less than $1$.

Using Inequalities to Compare Numbers

We can use inequalities to compare the numbers in the list. For example, we can use the following inequality to compare $\sqrt{0.8}$ and $\frac{4}{5}$:

• $\sqrt{0.8}$ is less than $\frac{4}{5}$ because $\sqrt{0.8}$ is less than $1$ and $\frac{4}{5}$ is greater than $1$.

Ordering the Numbers

Now that we have compared the numbers, we can order them from least to greatest:

1. $\sqrt{7} \cdot \frac{4}{5}$
2. $\sqrt{0.8}$
3. $\sqrt{7^2}$
4. $(\sqrt{5})^2$

Conclusion

In conclusion, Jana can use the following statement to create her list:

• $\sqrt{0.8}$ is less than $\frac{4}{5}$ because $\sqrt{0.8}$ is less than $1$ and $\frac{4}{5}$ is greater than $1$.

This statement allows Jana to determine the correct order of the numbers in the list.

Additional Examples

Here are some additional examples that demonstrate how to use inequalities to compare numbers:

• $\sqrt{2}$ is less than $2$ because $\sqrt{2}$ is less than $1$ and $1$ is less than $2$.
• $\sqrt{3}$ is greater than $2$ because $\sqrt{3}$ is greater than $1$ and $1$ is less than $2$.

Real-World Applications

The concept of ordering numbers is essential in many real-world applications, such as:

• Data analysis: When analyzing data, it is often necessary to order numbers from least to greatest or greatest to least.
• Statistics: In statistics, it is common to order numbers from least to greatest or greatest to least to calculate measures of central tendency and dispersion.
• Finance: In finance, it is often necessary to order numbers from least to greatest or greatest to least to compare investment returns or calculate interest rates.

Final Thoughts

Q: What is the correct order of the numbers in the list: $\sqrt{7^2}, \sqrt{7} \cdot \frac{4}{5}, \sqrt{0.8}, (\sqrt{5})^2$?

A: The correct order of the numbers in the list is:

1. $\sqrt{7} \cdot \frac{4}{5}$
2. $\sqrt{0.8}$
3. $\sqrt{7^2}$
4. $(\sqrt{5})^2$

Q: How do I determine the correct order of numbers?

A: To determine the correct order of numbers, you can use inequalities to compare them. For example, you can compare the square root of a number to 1, or compare two numbers that are multiplied together.

Q: What is the difference between $\sqrt{7^2}$ and $\sqrt{7}$?

A: $\sqrt{7^2}$ is equal to $7$, while $\sqrt{7}$ is less than $7$. This is because the square root of a number is always less than or equal to the number itself.

Q: How do I compare $\sqrt{0.8}$ and $\frac{4}{5}$?

A: To compare $\sqrt{0.8}$ and $\frac{4}{5}$, you can use the fact that $\sqrt{0.8}$ is less than 1 and $\frac{4}{5}$ is greater than 1. Therefore, $\sqrt{0.8}$ is less than $\frac{4}{5}$.

Q: What is the correct order of the numbers in the list: $\sqrt{2}, \sqrt{3}, \sqrt{4}$?

A: The correct order of the numbers in the list is:

1. $\sqrt{2}$
2. $\sqrt{3}$
3. $\sqrt{4}$

Q: How do I determine the correct order of numbers when they are multiplied together?

A: To determine the correct order of numbers when they are multiplied together, you can use the fact that the product of two numbers is always less than or equal to the product of the two numbers when they are multiplied together.

Q: What is the correct order of the numbers in the list: $\sqrt{5}, \sqrt{6}, \sqrt{7}$?

A: The correct order of the numbers in the list is:

1. $\sqrt{5}$
2. $\sqrt{6}$
3. $\sqrt{7}$

Q: How do I compare $\sqrt{9}$ and $\sqrt{10}$?

A: To compare $\sqrt{9}$ and $\sqrt{10}$, you can use the fact that $\sqrt{9}$ is equal to 3 and $\sqrt{10}$ is greater than 3. Therefore, $\sqrt{9}$ is less than $\sqrt{10}$.

Q: What is the correct order of the numbers in the list: $\sqrt{11}, \sqrt{12}, \sqrt{13}$?

A: The correct order of the numbers in the list is:

1. $\sqrt{11}$
2. $\sqrt{12}$
3. $\sqrt{13}$

Q: How do I determine the correct order of numbers when they are squared?

A: To determine the correct order of numbers when they are squared, you can use the fact that the square of a number is always greater than or equal to the number itself.

Q: What is the correct order of the numbers in the list: $\sqrt{14}, \sqrt{15}, \sqrt{16}$?

A: The correct order of the numbers in the list is:

1. $\sqrt{14}$
2. $\sqrt{15}$
3. $\sqrt{16}$

Conclusion

In conclusion, ordering numbers is a fundamental concept in mathematics that has many real-world applications. By using inequalities to compare numbers, we can determine the correct order of numbers from least to greatest or greatest to least. This concept is essential in many fields, including data analysis, statistics, and finance.