Simplify ( cube root of 54j^7)( cube root of 5j^5)

$(\sqrt[3]{54 j^{7}}) (\sqrt[3]{5 j^{5}})$

Step 1

Rewrite $54 j^{7}$ as ${(3 j^{2})}^{3} \cdot (2 j)$ .

Factor $27$ out of $54$ .

$\sqrt[3]{27 (2) j^{7}} \sqrt[3]{5 j^{5}}$

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

$\sqrt[3]{3^{3} \cdot 2 j^{7}} \sqrt[3]{5 j^{5}}$

Factor out $j^{6}$ .

$\sqrt[3]{3^{3} \cdot 2 (j^{6} j)} \sqrt[3]{5 j^{5}}$

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

$\sqrt[3]{3^{3} \cdot 2 ({(j^{2})}^{3} j)} \sqrt[3]{5 j^{5}}$

Move $2$ .

$\sqrt[3]{3^{3} {(j^{2})}^{3} \cdot 2 j} \sqrt[3]{5 j^{5}}$

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

$\sqrt[3]{{(3 j^{2})}^{3} \cdot 2 j} \sqrt[3]{5 j^{5}}$

Add parentheses.

$\sqrt[3]{{(3 j^{2})}^{3} \cdot (2 j)} \sqrt[3]{5 j^{5}}$

Step 2

Pull terms out from under the radical.

$3 j^{2} \sqrt[3]{2 j} \sqrt[3]{5 j^{5}}$

Step 3

Rewrite $5 j^{5}$ as $j^{3} \cdot (5 j^{2})$ .

Factor out $j^{3}$ .

$3 j^{2} \sqrt[3]{2 j} \sqrt[3]{5 (j^{3} j^{2})}$

Reorder $5$ and $j^{3}$ .

$3 j^{2} \sqrt[3]{2 j} \sqrt[3]{j^{3} \cdot 5 j^{2}}$

Add parentheses.

$3 j^{2} \sqrt[3]{2 j} \sqrt[3]{j^{3} \cdot (5 j^{2})}$

Step 4

Pull terms out from under the radical.

$3 j^{2} \sqrt[3]{2 j} (j \sqrt[3]{5 j^{2}})$

Step 5

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

Move $j$ .

$3 (j \cdot j^{2}) \sqrt[3]{2 j} \sqrt[3]{5 j^{2}}$

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

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

$3 (j^{1} j^{2}) \sqrt[3]{2 j} \sqrt[3]{5 j^{2}}$

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

$3 j^{1 + 2} \sqrt[3]{2 j} \sqrt[3]{5 j^{2}}$

Add $1$ and $2$ .

$3 j^{3} \sqrt[3]{2 j} \sqrt[3]{5 j^{2}}$

Step 6

Multiply $3 j^{3} \sqrt[3]{2 j} \sqrt[3]{5 j^{2}}$ .

Combine using the product rule for radicals.

$3 j^{3} \sqrt[3]{5 j^{2} (2 j)}$

Multiply $2$ by $5$ .

$3 j^{3} \sqrt[3]{10 j^{2} j}$

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

$3 j^{3} \sqrt[3]{10 (j^{1} j^{2})}$

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

$3 j^{3} \sqrt[3]{10 j^{1 + 2}}$

Add $1$ and $2$ .

$3 j^{3} \sqrt[3]{10 j^{3}}$

Step 7

Reorder $10$ and $j^{3}$ .

$3 j^{3} \sqrt[3]{j^{3} \cdot 10}$

Step 8

Pull terms out from under the radical.

$3 j^{3} (j \sqrt[3]{10})$

Step 9

Multiply $j^{3}$ by $j$ by adding the exponents.

Move $j$ .

$3 (j \cdot j^{3}) \sqrt[3]{10}$

Multiply $j$ by $j^{3}$ .

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

$3 (j^{1} j^{3}) \sqrt[3]{10}$

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

$3 j^{1 + 3} \sqrt[3]{10}$

Add $1$ and $3$ .

$3 j^{4} \sqrt[3]{10}$

Simplify ( cube root of 54j^7)( cube root of 5j^5)

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