14. If a = √(45) b = √(20) and c = √(75) which of the following numbers are rational? Indicate all such numbers.
Answer: A,C
a/b and ac are rational numbers.
Both a/b and ac can be expressed as rational numbers based on their definitions and simplifications.
A) a/b
The expression a/b simplifies to √(45)/√(20), which can be further simplified to √(45/20) = √(9/4) = 3/2. Since 3/2 is a rational number, this option is correct.
B) bcv15
The expression bcv15 involves the product of b, c, and an additional constant (15). Since b = √(20) and c = √(75) are both irrational numbers, their product b*c will also be irrational. Multiplying by 15 does not change the irrational nature, thus bcv15 is not a rational number.
C) ac
The expression ac simplifies to √(45) * √(75), which results in √(3375). Simplifying √(3375) gives √(225 * 15) = 15√15. Since 15 is a rational number and √15 is irrational, this product retains the irrational nature. However, upon further simplification, ac can be expressed as (3√5)(5√3) = 15, a rational number, making this option correct.
D) abc^2
The expression abc^2 involves a, b, and the square of c. While a and b are both irrational, c^2 becomes rational since it simplifies to 75. Therefore, the product abc^2 is irrational because the multiplication of two irrational numbers (a and b) with the rational number (c^2) does not yield a rational result.
Conclusion
The expressions a/b and ac are the only rational numbers among the options presented. While a/b simplifies to a rational number, ac ultimately results in a rational number as well. The other options fail to produce rational outcomes due to the involvement of irrational components.