Pre-Algebra Examples

$\sqrt[3]{25 a^{4 b}} (\sqrt[3]{40 a^{3 b^{2}}})$

Step 1

Rewrite $40 a^{3 b^{2}}$ as ${(2 a^{b^{2}})}^{3} \cdot 5$ .

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Step 1.1

Factor $8$ out of $40$ .

$\sqrt[3]{25 a^{4 b}} \sqrt[3]{8 (5) a^{3 b^{2}}}$

Step 1.2

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

$\sqrt[3]{25 a^{4 b}} \sqrt[3]{2^{3} \cdot 5 a^{3 b^{2}}}$

Step 1.3

Rewrite $a^{3 b^{2}}$ as ${(a^{b^{2}})}^{3}$ .

$\sqrt[3]{25 a^{4 b}} \sqrt[3]{2^{3} \cdot 5 {(a^{b^{2}})}^{3}}$

Step 1.4

Move $5$ .

$\sqrt[3]{25 a^{4 b}} \sqrt[3]{2^{3} {(a^{b^{2}})}^{3} \cdot 5}$

Step 1.5

Rewrite $2^{3} {(a^{b^{2}})}^{3}$ as ${(2 a^{b^{2}})}^{3}$ .

$\sqrt[3]{25 a^{4 b}} \sqrt[3]{{(2 a^{b^{2}})}^{3} \cdot 5}$

Step 2

Pull terms out from under the radical.

$\sqrt[3]{25 a^{4 b}} (2 a^{b^{2}} \sqrt[3]{5})$

Step 3

Rewrite using the commutative property of multiplication.

$2 \sqrt[3]{5} \sqrt[3]{25 a^{4 b}} a^{b^{2}}$

Step 4

Multiply $2 \sqrt[3]{5} \sqrt[3]{25 a^{4 b}}$ .

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Step 4.1

Combine using the product rule for radicals.

$2 \sqrt[3]{25 a^{4 b} \cdot 5} a^{b^{2}}$

Step 4.2

Multiply $5$ by $25$ .

$2 \sqrt[3]{125 a^{4 b}} a^{b^{2}}$

Step 5

Rewrite $125$ as $5^{3}$ .

$2 \sqrt[3]{5^{3} a^{4 b}} a^{b^{2}}$

Step 6

Pull terms out from under the radical.

$2 (5 \sqrt[3]{a^{4 b}}) a^{b^{2}}$

Step 7

Multiply $5$ by $2$ .

$10 \sqrt[3]{a^{4 b}} a^{b^{2}}$