Question

which statement is true about the expressions below?$sqrt{125} cdot 4.1overline{3}$$\frac{7}{9}+5sqrt{6}$$2.1 - \frac{sqrt3{81}}{3}$$sqrt3{-1}+sqrt{24}$a. all of the expressions result in a terminating decimal.b. all of the expressions are in simplest form.c. all of the expressions result in a rational number.d. all of the expressions result in an irrational number.

Explanation:

Step1: Simplify first expression

$\sqrt{125} \cdot 4.\overline{13} = 5\sqrt{5} \cdot 4.\overline{13}$
$\sqrt{5}$ is irrational; product of irrational and rational is irrational.

Step2: Simplify second expression

$\frac{7}{9} + 5\sqrt{6}$
$\sqrt{6}$ is irrational; sum of rational and irrational is irrational.

Step3: Simplify third expression

$2.1 - \frac{\sqrt[3]{81}}{3} = 2.1 - \frac{3\sqrt[3]{3}}{3} = 2.1 - \sqrt[3]{3}$
$\sqrt[3]{3}$ is irrational; difference of rational and irrational is irrational.

Step4: Simplify fourth expression

$\sqrt[3]{-1} + \sqrt{24} = -1 + 2\sqrt{6}$
$\sqrt{6}$ is irrational; sum of rational and irrational is irrational.

Step5: Evaluate options

A: False (no terminating decimals, all have irrationals)
B: False (first, third, fourth can be simplified)
C: False (all results are irrational)
D: True (all expressions result in irrational numbers)

Answer:

D. All of the expressions result in an irrational number.