Determine If $\sqrt{81}$ Is Rational Or Irrational And Give A Reason For Your Answer.

Determine if $\sqrt{81}$ is Rational or Irrational: A Mathematical Analysis

In mathematics, the concept of rational and irrational numbers is a fundamental aspect of understanding the properties of numbers. A rational number is a number that can be expressed as the quotient or fraction of two integers, where the denominator is non-zero. On the other hand, an irrational number is a number that cannot be expressed as a quotient of two integers. In this article, we will determine whether $\sqrt{81}$ is a rational or irrational number and provide a reason for our answer.

What is a Rational Number?

A rational number is a number that can be expressed as the quotient of two integers, where the denominator is non-zero. In other words, a rational number is a number that can be written in the form $\frac{a}{b}$, where $a$ and $b$ are integers and $b$ is non-zero. For example, the numbers $2$, $3$, and $\frac{1}{2}$ are all rational numbers because they can be expressed as the quotient of two integers.

What is an Irrational Number?

An irrational number is a number that cannot be expressed as a quotient of two integers. In other words, an irrational number is a number that cannot be written in the form $\frac{a}{b}$, where $a$ and $b$ are integers and $b$ is non-zero. For example, the numbers $\sqrt{2}$ and $\pi$ are both irrational numbers because they cannot be expressed as the quotient of two integers.

Is $\sqrt{81}$ a Rational or Irrational Number?

To determine whether $\sqrt{81}$ is a rational or irrational number, we need to examine its properties. The number $\sqrt{81}$ is a square root of the number $81$. We can simplify this expression by finding the prime factorization of $81$. The prime factorization of $81$ is $3^4$. Therefore, we can write $\sqrt{81}$ as $\sqrt{3^4}$.

Simplifying $\sqrt{3^4}$

We can simplify the expression $\sqrt{3^4}$ by using the property of exponents that states $\sqrt{a^n} = a^{\frac{n}{2}}$. Applying this property to the expression $\sqrt{3^4}$, we get:

$$\sqrt{3^4} = 3^{\frac{4}{2}} = 3^2 = 9$$

In conclusion, we have determined that $\sqrt{81}$ is a rational number. This is because we were able to simplify the expression $\sqrt{81}$ to the rational number $9$. Therefore, we can conclude that $\sqrt{81}$ is a rational number.

Why is $\sqrt{81}$ a Rational Number?

$\sqrt{81}$ is a rational number because it can be expressed as the quotient of two integers. In this case, we can write $\sqrt{81}$ as $\frac{9}{1}$, where $9$ and $1$ are integers. Therefore, $\sqrt{81}$ meets the definition of a rational number.

Real-World Applications of Rational Numbers

Rational numbers have many real-world applications. For example, rational numbers are used in finance to calculate interest rates and investment returns. They are also used in science to calculate the speed and distance of objects. In addition, rational numbers are used in engineering to design and build structures such as bridges and buildings.

In conclusion, we have determined that $\sqrt{81}$ is a rational number. This is because we were able to simplify the expression $\sqrt{81}$ to the rational number $9$. Therefore, we can conclude that $\sqrt{81}$ is a rational number. Rational numbers have many real-world applications and are an important concept in mathematics.

In our previous article, we discussed the concept of rational and irrational numbers and determined that $\sqrt{81}$ is a rational number. In this article, we will answer some frequently asked questions about rational and irrational numbers.

Q: What is the difference between a rational number and an irrational number?

A: A rational number is a number that can be expressed as the quotient of two integers, where the denominator is non-zero. An irrational number is a number that cannot be expressed as a quotient of two integers.

Q: Can all rational numbers be expressed as a decimal?

A: Yes, all rational numbers can be expressed as a decimal. For example, the rational number $\frac{1}{2}$ can be expressed as the decimal $0.5$.

Q: Can all irrational numbers be expressed as a decimal?

A: No, not all irrational numbers can be expressed as a decimal. For example, the irrational number $\sqrt{2}$ cannot be expressed as a decimal.

Q: How do we know if a number is rational or irrational?

A: To determine if a number is rational or irrational, we need to examine its properties. If the number can be expressed as the quotient of two integers, then it is a rational number. If the number cannot be expressed as the quotient of two integers, then it is an irrational number.

Q: Can we convert an irrational number to a rational number?

A: No, we cannot convert an irrational number to a rational number. Irrational numbers are unique and cannot be expressed as a quotient of two integers.

Q: What are some examples of rational numbers?

A: Some examples of rational numbers include:

• Whole numbers: 0, 1, 2, 3, ...
• Fractions: $\frac{1}{2}$, $\frac{3}{4}$, $\frac{5}{6}$, ...
• Decimals: 0.5, 0.25, 0.75, ...

Q: What are some examples of irrational numbers?

A: Some examples of irrational numbers include:

• Square roots: $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$, ...
• Pi: $\pi$
• Euler's number: $e$

Q: Can we add, subtract, multiply, and divide rational numbers?

A: Yes, we can add, subtract, multiply, and divide rational numbers. For example:

• $\frac{1}{2} + \frac{1}{3} = \frac{5}{6}$
• $\frac{1}{2} - \frac{1}{3} = \frac{1}{6}$
• $\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$
• $\frac{1}{2} \div \frac{1}{3} = \frac{3}{2}$

Q: Can we add, subtract, multiply, and divide irrational numbers?

A: No, we cannot add, subtract, multiply, and divide irrational numbers in the same way that we can with rational numbers. However, we can add and subtract irrational numbers, but the result may not be an irrational number.

In conclusion, we have answered some frequently asked questions about rational and irrational numbers. We hope that this article has provided a clear understanding of the concept of rational and irrational numbers and their properties.

Rational and irrational numbers are fundamental concepts in mathematics and have many real-world applications. Understanding the properties of rational and irrational numbers is essential for solving mathematical problems and making informed decisions in various fields. We hope that this article has provided a useful resource for students and professionals alike.