Simplify ( cube root of 2t^6)( cube root of 6t^8)

$(\sqrt[3]{2 t^{6}}) (\sqrt[3]{6 t^{8}})$

Step 1

Rewrite $2 t^{6}$ as ${(t^{2})}^{3} \cdot 2$ .

Rewrite $t^{6}$ as ${(t^{2})}^{3}$ .

$\sqrt[3]{2 {(t^{2})}^{3}} \sqrt[3]{6 t^{8}}$

Reorder $2$ and ${(t^{2})}^{3}$ .

$\sqrt[3]{{(t^{2})}^{3} \cdot 2} \sqrt[3]{6 t^{8}}$

Step 2

Pull terms out from under the radical.

$t^{2} \sqrt[3]{2} \sqrt[3]{6 t^{8}}$

Step 3

Rewrite $6 t^{8}$ as ${(t^{2})}^{3} \cdot (6 t^{2})$ .

Factor out $t^{6}$ .

$t^{2} \sqrt[3]{2} \sqrt[3]{6 (t^{6} t^{2})}$

Rewrite $t^{6}$ as ${(t^{2})}^{3}$ .

$t^{2} \sqrt[3]{2} \sqrt[3]{6 ({(t^{2})}^{3} t^{2})}$

Reorder $6$ and ${(t^{2})}^{3}$ .

$t^{2} \sqrt[3]{2} \sqrt[3]{{(t^{2})}^{3} \cdot 6 t^{2}}$

Add parentheses.

$t^{2} \sqrt[3]{2} \sqrt[3]{{(t^{2})}^{3} \cdot (6 t^{2})}$

Step 4

Pull terms out from under the radical.

$t^{2} \sqrt[3]{2} (t^{2} \sqrt[3]{6 t^{2}})$

Step 5

Multiply $t^{2}$ by $t^{2}$ by adding the exponents.

Move $t^{2}$ .

$t^{2} t^{2} \sqrt[3]{2} \sqrt[3]{6 t^{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$t^{2 + 2} \sqrt[3]{2} \sqrt[3]{6 t^{2}}$

Add $2$ and $2$ .

$t^{4} \sqrt[3]{2} \sqrt[3]{6 t^{2}}$

Step 6

Multiply $t^{4} \sqrt[3]{2} \sqrt[3]{6 t^{2}}$ .

Combine using the product rule for radicals.

$t^{4} \sqrt[3]{6 t^{2} \cdot 2}$

Multiply $2$ by $6$ .

$t^{4} \sqrt[3]{12 t^{2}}$