Simplify cube root of 4x^2 cube root of 12x^3 cube root of 36x^4

$\sqrt[3]{4 x^{2}} \sqrt[3]{12 x^{3}} \sqrt[3]{36 x^{4}}$

Step 1

Reorder $12$ and $x^{3}$ .

$\sqrt[3]{4 x^{2}} \sqrt[3]{x^{3} \cdot 12} \sqrt[3]{36 x^{4}}$

Step 2

Pull terms out from under the radical.

$\sqrt[3]{4 x^{2}} (x \sqrt[3]{12}) \sqrt[3]{36 x^{4}}$

Step 3

Rewrite using the commutative property of multiplication.

$\sqrt[3]{12} \sqrt[3]{4 x^{2}} x \sqrt[3]{36 x^{4}}$

Step 4

Multiply $\sqrt[3]{12} \sqrt[3]{4 x^{2}}$ .

Combine using the product rule for radicals.

$\sqrt[3]{12 (4 x^{2})} x \sqrt[3]{36 x^{4}}$

Multiply $12$ by $4$ .

$\sqrt[3]{48 x^{2}} x \sqrt[3]{36 x^{4}}$

Step 5

Rewrite $48 x^{2}$ as $2^{3} \cdot (6 x^{2})$ .

Factor $8$ out of $48$ .

$\sqrt[3]{8 (6) x^{2}} x \sqrt[3]{36 x^{4}}$

Rewrite $8$ as $2^{3}$ .

$\sqrt[3]{2^{3} \cdot 6 x^{2}} x \sqrt[3]{36 x^{4}}$

Add parentheses.

$\sqrt[3]{2^{3} \cdot (6 x^{2})} x \sqrt[3]{36 x^{4}}$

Step 6

Pull terms out from under the radical.

$2 \sqrt[3]{6 x^{2}} x \sqrt[3]{36 x^{4}}$

Step 7

Rewrite $36 x^{4}$ as $x^{3} \cdot (36 x)$ .

Factor out $x^{3}$ .

$2 \sqrt[3]{6 x^{2}} x \sqrt[3]{36 (x^{3} x)}$

Reorder $36$ and $x^{3}$ .

$2 \sqrt[3]{6 x^{2}} x \sqrt[3]{x^{3} \cdot 36 x}$

Add parentheses.

$2 \sqrt[3]{6 x^{2}} x \sqrt[3]{x^{3} \cdot (36 x)}$

Step 8

Pull terms out from under the radical.

$2 \sqrt[3]{6 x^{2}} x (x \sqrt[3]{36 x})$

Step 9

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

Move $x$ .

$2 \sqrt[3]{6 x^{2}} (x \cdot x) \sqrt[3]{36 x}$

Multiply $x$ by $x$ .

$2 \sqrt[3]{6 x^{2}} x^{2} \sqrt[3]{36 x}$

Step 10

Multiply $2 \sqrt[3]{6 x^{2}} x^{2} \sqrt[3]{36 x}$ .

Combine using the product rule for radicals.

$2 x^{2} \sqrt[3]{6 x^{2} (36 x)}$

Multiply $36$ by $6$ .

$2 x^{2} \sqrt[3]{216 x^{2} x}$

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

$2 x^{2} \sqrt[3]{216 (x^{1} x^{2})}$

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

$2 x^{2} \sqrt[3]{216 x^{1 + 2}}$

Add $1$ and $2$ .

$2 x^{2} \sqrt[3]{216 x^{3}}$

Step 11

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

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

Step 12

Pull terms out from under the radical, assuming real numbers.

$2 x^{2} (6 x)$

Step 13

Rewrite using the commutative property of multiplication.

$2 \cdot 6 x^{2} x$

Step 14

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

Move $x$ .

$2 \cdot 6 (x \cdot x^{2})$

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

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

$2 \cdot 6 (x^{1} x^{2})$

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

$2 \cdot 6 x^{1 + 2}$

Add $1$ and $2$ .

$2 \cdot 6 x^{3}$

Step 15

Multiply $2$ by $6$ .

$12 x^{3}$

Simplify cube root of 4x^2 cube root of 12x^3 cube root of 36x^4

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