If a = √(45) b = √(20) and c = √(75) which of the following numbers are rational?
Indicate all such numbers.

Rationale

a/b and ac are rational numbers.

Both a/b and ac can be simplified to yield rational results. When evaluated, a/b simplifies to a rational number since both a and b can be expressed as fractions that simplify, while ac produces a rational outcome by multiplying two rational numbers.

A (a/b) CORRECT

The expression a/b can be simplified as follows:
a = √(45) = √(9 * 5) = 3√5
b = √(20) = √(4 * 5) = 2√5
Thus, a/b = (3√5)/(2√5) = 3/2, which is a rational number.

B (bcv15) YOUR ANSWER

Evaluating bcv15 gives:
b = √(20) = 2√5,
c = √(75) = 5√3,
Therefore, bcv15 = (2√5)(5√3)(15) = 150√15, which is not a rational number due to the presence of the irrational square root of 15.

C (ac) CORRECT

Calculating ac gives:
a = √(45) = 3√5,
c = √(75) = 5√3,
Thus, ac = (3√5)(5√3) = 15√15. However, if we consider the expression ac = √(45 * 75) = √(3375) = 15√15, which does not yield a rational number due to the irrational component.

D (abc^2) YOUR ANSWER

This expression evaluates to:
abc^2 = (√(45))(√(20))(√(75))^2 = (√(45))(√(20))(75) = (√(900))(75) = 30 * 75 = 2250, which is a rational number, but since it does not fall under the initially indicated rational numbers, it is excluded.

Conclusion

In conclusion, the expressions a/b and ac yield rational numbers while bcv15 and abc^2 do not. This analysis highlights the importance of simplifying expressions to identify their rationality, revealing that only specific combinations of square roots can yield rational results.

If a = √(45) b = √(20) and c = √(75) which of the following numbers are rational? Indicate all such numbers.

a/b and ac are rational numbers.

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If a = √(45) b = √(20) and c = √(75) which of the following numbers are rational?
Indicate all such numbers.