Select All The Expressions That Are Rational Numbers.A. \[$(3 \sqrt{5}) + (2 \sqrt{5})\$\]B. \[$(3 \sqrt{5}) - (2 \sqrt{5})\$\]C. \[$(3 \sqrt{5}) \times (2 \sqrt{5})\$\]D. \[$(3 \sqrt{5}) \div (2 \sqrt{5})\$\]E.

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Introduction

Rational numbers are a fundamental concept in mathematics, and understanding them is crucial for solving various mathematical problems. In this article, we will delve into the concept of rational numbers, explore their properties, and learn how to identify them. We will also examine a set of expressions and determine which ones represent rational numbers.

What are Rational Numbers?

Rational numbers are numbers that can be expressed as the ratio of two integers, i.e., they can be written in the form a/b, where a and b are integers and b is non-zero. Rational numbers include all integers, fractions, and decimals that can be expressed as a finite decimal or fraction. Examples of rational numbers include 3, 1/2, 0.5, and 22/7.

Properties of Rational Numbers

Rational numbers have several properties that make them unique. Some of the key properties include:

  • Closure: Rational numbers are closed under addition, subtraction, multiplication, and division (except by zero).
  • Commutativity: Rational numbers are commutative under addition and multiplication.
  • Associativity: Rational numbers are associative under addition and multiplication.
  • Distributivity: Rational numbers are distributive under multiplication over addition.

Identifying Rational Numbers

To identify rational numbers, we need to look for expressions that can be written in the form a/b, where a and b are integers and b is non-zero. Let's examine the given expressions and determine which ones represent rational numbers.

Expression A: {(3 \sqrt{5}) + (2 \sqrt{5})$}$

This expression involves the addition of two square roots. Since the square roots are not simplified, the expression cannot be written in the form a/b, where a and b are integers. Therefore, expression A is not a rational number.

Expression B: {(3 \sqrt{5}) - (2 \sqrt{5})$}$

This expression involves the subtraction of two square roots. Similar to expression A, the square roots are not simplified, and the expression cannot be written in the form a/b, where a and b are integers. Therefore, expression B is not a rational number.

Expression C: {(3 \sqrt{5}) \times (2 \sqrt{5})$}$

This expression involves the multiplication of two square roots. When we multiply the square roots, we get:

{(3 \sqrt{5}) \times (2 \sqrt{5}) = 6 \times 5 = 30$}$

Since the result is an integer, expression C can be written in the form a/b, where a and b are integers. Therefore, expression C is a rational number.

Expression D: {(3 \sqrt{5}) \div (2 \sqrt{5})$}$

This expression involves the division of two square roots. When we divide the square roots, we get:

{(3 \sqrt{5}) \div (2 \sqrt{5}) = \frac{3}{2}$}$

Since the result is a fraction, expression D can be written in the form a/b, where a and b are integers. Therefore, expression D is a rational number.

Expression E: {\frac{1}{\sqrt{5}}$}$

This expression involves the division of 1 by a square root. When we simplify the expression, we get:

{\frac{1}{\sqrt{5}} = \frac{1}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{5}}{5}$}$

Since the result is a fraction, expression E can be written in the form a/b, where a and b are integers. Therefore, expression E is a rational number.

Conclusion

In conclusion, rational numbers are numbers that can be expressed as the ratio of two integers. They have several properties, including closure, commutativity, associativity, and distributivity. To identify rational numbers, we need to look for expressions that can be written in the form a/b, where a and b are integers and b is non-zero. In this article, we examined a set of expressions and determined which ones represent rational numbers. We found that expressions C, D, and E are rational numbers, while expressions A and B are not.

Final Answer

The correct expressions that are rational numbers are:

  • Expression C: {(3 \sqrt{5}) \times (2 \sqrt{5})$}$
  • Expression D: {(3 \sqrt{5}) \div (2 \sqrt{5})$}$
  • Expression E: {\frac{1}{\sqrt{5}}$}$
    Rational Numbers: Frequently Asked Questions =====================================================

Introduction

Rational numbers are a fundamental concept in mathematics, and understanding them is crucial for solving various mathematical problems. In this article, we will answer some frequently asked questions about rational numbers, providing a deeper understanding of this concept.

Q: What is the difference between rational and irrational numbers?

A: Rational numbers are numbers that can be expressed as the ratio of two integers, i.e., they can be written in the form a/b, where a and b are integers and b is non-zero. Irrational numbers, on the other hand, are numbers that cannot be expressed as a finite decimal or fraction. Examples of irrational numbers include π (pi) and the square root of 2.

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

A: Yes, all rational numbers can be expressed as a decimal. In fact, rational numbers can be expressed as a finite decimal or a repeating decimal. For example, the rational number 1/2 can be expressed as 0.5, while the rational number 1/3 can be expressed as 0.333... (where the dots indicate that the 3's repeat infinitely).

Q: Can all decimals be expressed as a rational number?

A: No, not all decimals can be expressed as a rational number. For example, the decimal 0.123456789101112... (where the numbers 1 through 12 repeat infinitely) cannot be expressed as a rational number. This is because it is a non-repeating, non-terminating decimal, which is a characteristic of irrational numbers.

Q: How do I determine if a number is rational or irrational?

A: To determine if a number is rational or irrational, you can try to express it as a fraction or a decimal. If it can be expressed as a fraction or a decimal, it is likely a rational number. If it cannot be expressed as a fraction or a decimal, it is likely an irrational number.

Q: Can rational numbers be negative?

A: Yes, rational numbers can be negative. For example, the rational number -3/4 is a negative rational number.

Q: Can rational numbers be fractions with a negative denominator?

A: Yes, rational numbers can be fractions with a negative denominator. For example, the rational number 3/-4 is a fraction with a negative denominator.

Q: Can rational numbers be expressed as a percentage?

A: Yes, rational numbers can be expressed as a percentage. For example, the rational number 1/2 can be expressed as 50%.

Q: Can rational numbers be used in real-world applications?

A: Yes, rational numbers are used in many real-world applications, including finance, science, and engineering. For example, the price of a stock or a bond can be expressed as a rational number, and the interest rate on a loan can be expressed as a rational number.

Q: Can irrational numbers be used in real-world applications?

A: Yes, irrational numbers are also used in many real-world applications, including physics and engineering. For example, the speed of light is an irrational number, and the frequency of a wave can be an irrational number.

Conclusion

In conclusion, rational numbers are a fundamental concept in mathematics, and understanding them is crucial for solving various mathematical problems. We have answered some frequently asked questions about rational numbers, providing a deeper understanding of this concept. Whether you are a student, a teacher, or a professional, understanding rational numbers is essential for success in mathematics and beyond.

Final Answer

The answers to the frequently asked questions about rational numbers are:

  • Q: What is the difference between rational and irrational numbers? A: Rational numbers are numbers that can be expressed as the ratio of two integers, while irrational numbers are numbers that cannot be expressed as a finite decimal or fraction.
  • Q: Can all rational numbers be expressed as a decimal? A: Yes, all rational numbers can be expressed as a decimal.
  • Q: Can all decimals be expressed as a rational number? A: No, not all decimals can be expressed as a rational number.
  • Q: How do I determine if a number is rational or irrational? A: To determine if a number is rational or irrational, you can try to express it as a fraction or a decimal.
  • Q: Can rational numbers be negative? A: Yes, rational numbers can be negative.
  • Q: Can rational numbers be fractions with a negative denominator? A: Yes, rational numbers can be fractions with a negative denominator.
  • Q: Can rational numbers be expressed as a percentage? A: Yes, rational numbers can be expressed as a percentage.
  • Q: Can rational numbers be used in real-world applications? A: Yes, rational numbers are used in many real-world applications.
  • Q: Can irrational numbers be used in real-world applications? A: Yes, irrational numbers are also used in many real-world applications.