Simplify cube root of 216x^5* fourth root of 81x^7

$\sqrt[3]{216 x^{5}} \cdot \sqrt[4]{81 x^{7}}$

Step 1

Rewrite $216 x^{5}$ as ${(6 x)}^{3} x^{2}$ .

Rewrite $216$ as $6^{3}$ .

$\sqrt[3]{6^{3} x^{5}} \cdot \sqrt[4]{81 x^{7}}$

Factor out $x^{3}$ .

$\sqrt[3]{6^{3} (x^{3} x^{2})} \cdot \sqrt[4]{81 x^{7}}$

Rewrite $6^{3} x^{3}$ as ${(6 x)}^{3}$ .

$\sqrt[3]{{(6 x)}^{3} x^{2}} \cdot \sqrt[4]{81 x^{7}}$

Step 2

Pull terms out from under the radical.

$6 x \sqrt[3]{x^{2}} \cdot \sqrt[4]{81 x^{7}}$

Step 3

Rewrite $81 x^{7}$ as ${(3 x)}^{4} x^{3}$ .

Rewrite $81$ as $3^{4}$ .

$6 x \sqrt[3]{x^{2}} \cdot \sqrt[4]{3^{4} x^{7}}$

Factor out $x^{4}$ .

$6 x \sqrt[3]{x^{2}} \cdot \sqrt[4]{3^{4} (x^{4} x^{3})}$

Rewrite $3^{4} x^{4}$ as ${(3 x)}^{4}$ .

$6 x \sqrt[3]{x^{2}} \cdot \sqrt[4]{{(3 x)}^{4} x^{3}}$

Step 4

Pull terms out from under the radical.

$6 x \sqrt[3]{x^{2}} \cdot (3 x \sqrt[4]{x^{3}})$

Step 5

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$6 (x \cdot x) \sqrt[3]{x^{2}} \cdot (3 \sqrt[4]{x^{3}})$

Multiply $x$ by $x$ .

$6 x^{2} \sqrt[3]{x^{2}} \cdot (3 \sqrt[4]{x^{3}})$

Step 6

Multiply $6 x^{2} \sqrt[3]{x^{2}} (3 \sqrt[4]{x^{3}})$ .

Multiply $3$ by $6$ .

$18 x^{2} \sqrt[3]{x^{2}} \sqrt[4]{x^{3}}$

Rewrite the expression using the least common index of $12$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x^{3}}$ as $x^{\frac{3}{4}}$ .

$18 x^{2} (x^{\frac{3}{4}} \sqrt[3]{x^{2}})$

Rewrite $x^{\frac{3}{4}}$ as $x^{\frac{9}{12}}$ .

$18 x^{2} (x^{\frac{9}{12}} \sqrt[3]{x^{2}})$

Rewrite $x^{\frac{9}{12}}$ as $\sqrt[12]{x^{9}}$ .

$18 x^{2} (\sqrt[12]{x^{9}} \sqrt[3]{x^{2}})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

$18 x^{2} (\sqrt[12]{x^{9}} x^{\frac{2}{3}})$

Rewrite $x^{\frac{2}{3}}$ as $x^{\frac{8}{12}}$ .

$18 x^{2} (\sqrt[12]{x^{9}} x^{\frac{8}{12}})$

Rewrite $x^{\frac{8}{12}}$ as $\sqrt[12]{x^{8}}$ .

$18 x^{2} (\sqrt[12]{x^{9}} \sqrt[12]{x^{8}})$

Combine using the product rule for radicals.

$18 x^{2} \sqrt[12]{x^{9} x^{8}}$

Multiply $x^{9}$ by $x^{8}$ by adding the exponents.

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

$18 x^{2} \sqrt[12]{x^{9 + 8}}$

Add $9$ and $8$ .

$18 x^{2} \sqrt[12]{x^{17}}$

Step 7

Factor out $x^{12}$ .

$18 x^{2} \sqrt[12]{x^{12} x^{5}}$

Step 8

Pull terms out from under the radical.

$18 x^{2} (x \sqrt[12]{x^{5}})$

Step 9

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

Move $x$ .

$18 (x \cdot x^{2}) \sqrt[12]{x^{5}}$

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

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

$18 (x^{1} x^{2}) \sqrt[12]{x^{5}}$

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

$18 x^{1 + 2} \sqrt[12]{x^{5}}$

Add $1$ and $2$ .

$18 x^{3} \sqrt[12]{x^{5}}$

Simplify cube root of 216x^5* fourth root of 81x^7

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