Question

question
write the following expression in simplest radical form.
\\(\left(\sqrt3{1024q^4}\
ight)\left(\sqrt3{320q^2}\
ight)\\)

Explanation:

Step1: Combine the cube roots

$$\sqrt[3]{1024q^4 \cdot 320q^2}$$

Step2: Multiply coefficients and variables

$$\sqrt[3]{(1024 \times 320)q^{4+2}} = \sqrt[3]{327680q^6}$$

Step3: Factor into perfect cubes

$$\sqrt[3]{10240 \times 32 \times q^6} = \sqrt[3]{64 \times 5120 \times q^6} = \sqrt[3]{64 \times 64 \times 80 \times q^6} = \sqrt[3]{64^2 \times 80 \times q^6}$$
$$= \sqrt[3]{(4^3)^2 \times 80 \times (q^2)^3} = \sqrt[3]{4^6 \times 80 \times q^6}$$

Step4: Extract perfect cubes

$$4^2 q^2 \sqrt[3]{80} = 16q^2\sqrt[3]{80}$$

Step5: Simplify remaining radical

$$16q^2\sqrt[3]{16 \times 5} = 16q^2\sqrt[3]{8 \times 2 \times 5} = 16q^2 \times 2\sqrt[3]{10} = 32q^2\sqrt[3]{10}$$

Answer:

$32q^2\sqrt[3]{10}$