Question

what is the following quotient? \\(\frac{\sqrt3{60}}{\sqrt3{20}}\\)
options:
\\(2\left(\sqrt3{5}\
ight)\\)
\\(\sqrt3{3}\\)
40
3

Explanation:

Step1: Simplify the cube root fraction

We have \(\frac{\sqrt[3]{60}}{\sqrt[3]{20}}\). Using the property of cube roots \(\frac{\sqrt[3]{a}}{\sqrt[3]{b}}=\sqrt[3]{\frac{a}{b}}\), so this becomes \(\sqrt[3]{\frac{60}{20}}\).

Step2: Simplify the fraction inside the cube root

Calculate \(\frac{60}{20} = 3\)? Wait, no, wait: \(60\div20 = 3\)? Wait, no, 20 times 3 is 60? Wait, no, 20 times 3 is 60? Wait, 203=60, yes. Wait, but wait, maybe I made a mistake. Wait, 60 and 20, let's factor them. 60 = 203, so \(\frac{60}{20}=3\)? Wait, no, that can't be. Wait, no, 60 divided by 20 is 3? Wait, 20*3=60, yes. Wait, but then \(\sqrt[3]{3}\)? But wait, the options have \(\sqrt[3]{3}\) as one of them. Wait, let's check again.

Wait, \(\frac{\sqrt[3]{60}}{\sqrt[3]{20}}=\sqrt[3]{\frac{60}{20}}=\sqrt[3]{3}\). Wait, but let's factor 60 and 20. 60 = 2^2 3 5, 20 = 2^2 * 5. So \(\frac{60}{20}=\frac{2^2*3*5}{2^2*5}=3\). So the cube root of 3 is \(\sqrt[3]{3}\). Wait, but the option with \(\sqrt[3]{3}\) is one of the choices. Wait, let's check the options again. The options are: \(2(\sqrt[3]{5})\), \(\sqrt[3]{3}\), 40, 3. So according to the calculation, \(\sqrt[3]{3}\) is the result.

Wait, maybe I messed up the factoring. Wait, 60 is 20*3, so when we take the cube root of 60 over 20, it's cube root of 3. So the correct answer should be \(\sqrt[3]{3}\).

Answer:

\(\sqrt[3]{3}\) (the option with \(\sqrt[3]{3}\))