Question

Which expression result in a rational number? Select all that apply.

A = √3
B = 2√3
C= √25
D= √16
Choices
1. A+B
2. C+D
3. A+D
4. A x D
5. B x D
6. C x D
7. A x A

Answer 1

Expressions C + D (√25 + √16), C x D (√25 x √16), and A x A (√3 x √3) are the ones that result in rational numbers because they involve adding or multiplying square roots of perfect squares or squaring an irrational number.

Step-by-step explanation:

The student has asked which expressions result in a rational number. We are given several expressions involving square roots and are to determine which combinations of these will yield rational numbers. A rational number is a number that can be expressed as the quotient of two integers, and it has a finite or repeating decimal expansion.

Let's evaluate each option:

1. A + B: This is √3 + 2√3. As √3 is an irrational number, adding it to a multiple of itself still results in an irrational number.
2. C + D: This is √25 + √16. Both √25 and √16 are rational numbers (5 and 4 respectively), so their sum 5 + 4 = 9 is also a rational number.
3. A + D: This is √3 + √16. As √3 is irrational and √16 is rational (4), their sum is irrational.
4. A x D: This is √3 x √16. This simplifies to √3 x 4, which is irrational because √3 is multiplied by an integer, resulting in an irrational number.
5. B x D: This is 2√3 x √16. Simplifying, we get 2√3 x 4, which remains irrational for the same reason as the previous expression.
6. C x D: This is √25 x √16. Simplifying, we get 5 x 4, which is 20, a rational number.
7. A x A: This is √3 x √3, which simplifies to 3, a rational number.

From these evaluations, we conclude that the expressions C + D, C x D, and A x A result in rational numbers.