Question

write $sqrt3{\frac{64}{49}}$ as a radical expression in simplest form.
$sqrt3{\frac{64}{49}} = \frac{square}{square}$

Explanation:

Step1: Split root into fraction of roots

$$\sqrt[3]{\frac{64}{49}} = \frac{\sqrt[3]{64}}{\sqrt[3]{49}}$$

Step2: Simplify perfect cube root

$\sqrt[3]{64} = 4$, so $\frac{4}{\sqrt[3]{49}}$

Step3: Rationalize the denominator

Multiply numerator and denominator by $\sqrt[3]{49^2}$:
$$\frac{4 \cdot \sqrt[3]{49^2}}{\sqrt[3]{49} \cdot \sqrt[3]{49^2}} = \frac{4\sqrt[3]{2401}}{\sqrt[3]{49^3}}$$

Step4: Simplify denominator root

$\sqrt[3]{49^3} = 49$, so $\frac{4\sqrt[3]{2401}}{49}$

Step5: Simplify the radical

$\sqrt[3]{2401} = \sqrt[3]{49^2} = \sqrt[3]{(7^2)^2} = \sqrt[3]{7^4} = 7\sqrt[3]{7}$, substitute back:
$$\frac{4 \cdot 7\sqrt[3]{7}}{49} = \frac{28\sqrt[3]{7}}{49} = \frac{4\sqrt[3]{7}}{7}$$

Answer:

$\frac{4\sqrt[3]{7}}{7}$