What Is The Sum And Classification Of $\frac{2}{5} + \sqrt{88}$?A. $9.78083151 \ldots$, Irrational B. $9.78083151 \ldots$, Rational C. $13.38083151 \ldots$, Irrational D. $13.38083151 \ldots$, Rational

Introduction

In mathematics, the sum of two numbers can be a rational or irrational number, depending on the nature of the numbers involved. When we add a rational number and an irrational number, the result can be either rational or irrational. In this article, we will explore the sum and classification of $\frac{2}{5} + \sqrt{88}$.

Understanding Rational and Irrational Numbers

Before we proceed, let's briefly discuss rational and irrational numbers. A rational number is a number that can be expressed as the ratio of two integers, i.e., $\frac{p}{q}$, where $p$ and $q$ are integers and $q$ is non-zero. Examples of rational numbers include $\frac{1}{2}$, $\frac{3}{4}$, and $2$. On the other hand, an irrational number is a number that cannot be expressed as a ratio of two integers. Examples of irrational numbers include $\sqrt{2}$, $\pi$, and $e$.

The Sum of $\frac{2}{5}$ and $\sqrt{88}$

Now, let's focus on the sum of $\frac{2}{5}$ and $\sqrt{88}$. To add these two numbers, we need to find a common denominator. Since $\frac{2}{5}$ is already in its simplest form, we can leave it as is. However, $\sqrt{88}$ can be simplified by factoring the number under the square root sign.

Simplifying $\sqrt{88}$

To simplify $\sqrt{88}$, we can factor the number under the square root sign as follows:

$$\sqrt{88} = \sqrt{4 \cdot 22} = \sqrt{4} \cdot \sqrt{22} = 2\sqrt{22}$$

Adding $\frac{2}{5}$ and $2\sqrt{22}$

Now that we have simplified $\sqrt{88}$, we can add it to $\frac{2}{5}$:

$$\frac{2}{5} + 2\sqrt{22}$$

To add these two numbers, we need to find a common denominator. Since $\frac{2}{5}$ is already in its simplest form, we can leave it as is. However, $2\sqrt{22}$ can be written as $\frac{2\sqrt{22} \cdot 5}{5}$ to get a common denominator:

$$\frac{2}{5} + \frac{2\sqrt{22} \cdot 5}{5} = \frac{2 + 10\sqrt{22}}{5}$$

Evaluating the Sum

Now that we have added $\frac{2}{5}$ and $2\sqrt{22}$, we can evaluate the sum:

$$\frac{2 + 10\sqrt{22}}{5}$$

To evaluate this expression, we need to calculate the value of $10\sqrt{22}$. Since $\sqrt{22}$ is an irrational number, $10\sqrt{22}$ is also an irrational number. Therefore, the sum $\frac{2 + 10\sqrt{22}}{5}$ is also an irrational number.

Conclusion

In conclusion, the sum of $\frac{2}{5}$ and $\sqrt{88}$ is $\frac{2 + 10\sqrt{22}}{5}$. Since $10\sqrt{22}$ is an irrational number, the sum is also an irrational number. Therefore, the correct answer is:

• A. $9.78083151 \ldots$, irrational

Note that the value of the sum is approximately $9.78083151 \ldots$, but it is not a rational number.

Final Answer

The final answer is $\boxed{A. 9.78083151 \ldots, irrational}$

Introduction

In our previous article, we explored the sum and classification of $\frac{2}{5} + \sqrt{88}$. We found that the sum is an irrational number. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the nature of the number $\sqrt{88}$?

A: The number $\sqrt{88}$ is an irrational number because it cannot be expressed as a ratio of two integers.

Q: Can you simplify $\sqrt{88}$?

A: Yes, we can simplify $\sqrt{88}$ by factoring the number under the square root sign. We can write $\sqrt{88}$ as $\sqrt{4 \cdot 22} = \sqrt{4} \cdot \sqrt{22} = 2\sqrt{22}$.

Q: How do you add $\frac{2}{5}$ and $2\sqrt{22}$?

A: To add $\frac{2}{5}$ and $2\sqrt{22}$, we need to find a common denominator. Since $\frac{2}{5}$ is already in its simplest form, we can leave it as is. However, $2\sqrt{22}$ can be written as $\frac{2\sqrt{22} \cdot 5}{5}$ to get a common denominator. The sum is then $\frac{2 + 10\sqrt{22}}{5}$.

Q: Is the sum $\frac{2}{5} + \sqrt{88}$ rational or irrational?

A: The sum $\frac{2}{5} + \sqrt{88}$ is irrational because $10\sqrt{22}$ is an irrational number.

Q: Can you provide an approximate value of the sum $\frac{2}{5} + \sqrt{88}$?

A: Yes, we can provide an approximate value of the sum $\frac{2}{5} + \sqrt{88}$. The sum is approximately $9.78083151 \ldots$.

Q: Why is the sum $\frac{2}{5} + \sqrt{88}$ not a rational number?

A: The sum $\frac{2}{5} + \sqrt{88}$ is not a rational number because $10\sqrt{22}$ is an irrational number. Since the sum contains an irrational number, it is also an irrational number.

Q: Can you provide a final answer to the question?

A: Yes, the final answer to the question is:

• A. $9.78083151 \ldots$, irrational

Note that the value of the sum is approximately $9.78083151 \ldots$, but it is not a rational number.

Conclusion

In conclusion, we have answered some frequently asked questions related to the sum and classification of $\frac{2}{5} + \sqrt{88}$. We have found that the sum is an irrational number and provided an approximate value of the sum.

Final Answer

The final answer is $\boxed{A. 9.78083151 \ldots, irrational}$