Question

A solution to an equation Jada solved is −2+3⎯⎯√.

She is trying to determine whether that solution is rational or irrational. Jada knows that −2
is a rational number and 3⎯⎯√
is an irrational number. She also knows that the sum of any two rational numbers is always rational.



a. If we add 2
(a rational number) to −2+3⎯⎯√,
what is the sum? Is the sum rational or irrational?



b. Explain why −2+3⎯⎯√
cannot be rational.

Answer 1

a. To find the sum of 2 (a rational number) and (-2 + √3), we add them together:

2 + (-2 + √3) = 0 + √3 = √3

The sum is √3. Since √3 is an irrational number, the sum is irrational.

b. The expression -2 + √3 cannot be rational because a rational number can be expressed as the ratio of two integers, while √3 is an irrational number. If -2 + √3 were rational, it could be expressed as a fraction a/b, where a and b are integers. However, we know that √3 is irrational, so it cannot be written as a fraction of two integers. Therefore, -2 + √3 cannot be rational.