Choose which set or sets the following number belongs to. Be sure to account for ALL sets.
StartRoot 18 EndRoot
A. rational numbers, real numbers
B. irrational numbers
C. rational numbers
D. real numbers
E. irrational numbers, real numbers
The number √18 is a positive square root, so it belongs to both the set of irrational numbers and the set of real numbers (since all irrational numbers are also real numbers). Therefore, the correct answer is option E. irrational numbers, real numbers.
To determine which set or sets the number √18 belongs to, we need to consider its properties.
The square root of 18, √18, can be simplified as follows:
√18 = √(9 x 2) = √9 x √2 = 3 x √2
Since we are dealing with the square root of 18, it is important to consider the nature of the number 18 itself.
The number 18 is a composite number (not a prime number) since it can be factored into smaller whole numbers:
18 = 2 x 9
Both 2 and 9 are rational numbers. In fact, any whole number is considered a rational number because it can be expressed as a fraction with a denominator of 1.
Furthermore, since the square root of 2 (√2) is an irrational number, we can conclude the following:
√18 = 3 x √2 is a product of a rational number (3) and an irrational number (√2).
Therefore, the number √18 belongs to the sets:
A. rational numbers, real numbers (because 3 is a rational number and √2 is an irrational number)
B. irrational numbers (because √2 is an irrational number)
C. rational numbers (because 3 is a rational number)
D. real numbers (since all rational and irrational numbers are real numbers)
E. irrational numbers, real numbers (because 3 is a rational number and √2 is an irrational number)
So, the correct answer is option E: irrational numbers, real numbers.