Question

mr. munoz gave his students the following expression to simplify as their exit ticket for class one day.

√24 + 3.85

julian used a calculator to simplify the expression and determined that the expression simplifies to be a rational number. below is the work he showed on his exit ticket.

√24 + 3.85 = 4.89898 + 3.85 = 8.74898

determine if julian is correct and provide justification for that conclusion.

a. julian is incorrect because √24 results in a non - repeating, non - terminating decimal. this means it is an irrational number, and the sum will also result in an irrational number.
b. julian is correct because the result is a terminating decimal. this means that the sum is a rational number.
c. julian is correct because √24 results in a terminating decimal. this means it is a rational number, and the sum will also result in a rational number.
d. julian is incorrect because the result is a repeating decimal. this means that the sum is an irrational number.

Explanation:

• A rational number can be expressed as a fraction \(\frac{p}{q}\) where \(p,q\in\mathbb{Z}\) and \(q

eq0\). Terminating or repeating decimals are rational. Non - repeating, non - terminating decimals are irrational.

• \(\sqrt{24}=\sqrt{4\times6} = 2\sqrt{6}\approx4.89897948557\cdots\) is a non - repeating, non - terminating decimal (since \(\sqrt{6}\) is irrational). When we add an irrational number (\(\sqrt{24}\)) and a rational number (\(3.85=\frac{385}{100}=\frac{77}{20}\)), the sum of an irrational and a rational number is always irrational.

Answer:

A. Julian is incorrect because \(\sqrt{24}\) results in a non - repeating, non - terminating decimal. This means it is an irrational number, and the sum will also result in an irrational number.