Question

1. write the expression in simplest form. write your answer as a radical expression.

\\(\frac{\sqrt3{5}}{\sqrt3{4}} = \square\\)

Explanation:

Step1: Use the property of radicals \(\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}\)

$$\frac{\sqrt[3]{5}}{\sqrt[3]{4}}=\sqrt[3]{\frac{5}{4}}$$

Step2: Rationalize the denominator

Multiply numerator and denominator inside the cube - root by \(4^{2}\) (since \(4\times4^{2}=4^{3}\)).
$$\sqrt[3]{\frac{5}{4}}=\sqrt[3]{\frac{5\times4^{2}}{4\times4^{2}}}=\sqrt[3]{\frac{5\times16}{4^{3}}}$$

Step3: Simplify the expression

$$\sqrt[3]{\frac{80}{64}}=\frac{\sqrt[3]{80}}{\sqrt[3]{64}}$$
Since \(\sqrt[3]{64} = 4\) and \(80=16\times5\), we can write \(\sqrt[3]{80}=\sqrt[3]{16\times5}=\sqrt[3]{8\times2\times5}=2\sqrt[3]{10}\)

Answer:

\(\frac{\sqrt[3]{80}}{4}\)