Simplify ( cube root of 20t^3)( cube root of 81t^7)

$(\sqrt[3]{20 t^{3}}) (\sqrt[3]{81 t^{7}})$

Step 1

Reorder $20$ and $t^{3}$ .

$\sqrt[3]{t^{3} \cdot 20} \sqrt[3]{81 t^{7}}$

Step 2

Pull terms out from under the radical.

$t \sqrt[3]{20} \sqrt[3]{81 t^{7}}$

Step 3

Rewrite $81 t^{7}$ as ${(3 t^{2})}^{3} \cdot (3 t)$ .

Factor $27$ out of $81$ .

$t \sqrt[3]{20} \sqrt[3]{27 (3) t^{7}}$

Rewrite $27$ as $3^{3}$ .

$t \sqrt[3]{20} \sqrt[3]{3^{3} \cdot 3 t^{7}}$

Factor out $t^{6}$ .

$t \sqrt[3]{20} \sqrt[3]{3^{3} \cdot 3 (t^{6} t)}$

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

$t \sqrt[3]{20} \sqrt[3]{3^{3} \cdot 3 ({(t^{2})}^{3} t)}$

Move $3$ .

$t \sqrt[3]{20} \sqrt[3]{3^{3} {(t^{2})}^{3} \cdot 3 t}$

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

$t \sqrt[3]{20} \sqrt[3]{{(3 t^{2})}^{3} \cdot 3 t}$

Add parentheses.

$t \sqrt[3]{20} \sqrt[3]{{(3 t^{2})}^{3} \cdot (3 t)}$

Step 4

Pull terms out from under the radical.

$t \sqrt[3]{20} (3 t^{2} \sqrt[3]{3 t})$

Step 5

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

Move $t^{2}$ .

$t^{2} t \sqrt[3]{20} (3 \sqrt[3]{3 t})$

Multiply $t^{2}$ by $t$ .

Raise $t$ to the power of $1$ .

$t^{2} t^{1} \sqrt[3]{20} (3 \sqrt[3]{3 t})$

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

$t^{2 + 1} \sqrt[3]{20} (3 \sqrt[3]{3 t})$

Add $2$ and $1$ .

$t^{3} \sqrt[3]{20} (3 \sqrt[3]{3 t})$

Step 6

Multiply $t^{3} \sqrt[3]{20} (3 \sqrt[3]{3 t})$ .

Combine using the product rule for radicals.

$t^{3} (3 \sqrt[3]{20 (3 t)})$

Multiply $20$ by $3$ .

$t^{3} (3 \sqrt[3]{60 t})$

Step 7

Rewrite using the commutative property of multiplication.

$3 t^{3} \sqrt[3]{60 t}$

Simplify ( cube root of 20t^3)( cube root of 81t^7)

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