Simplify the Radical Expression square root of 2b^3 cube root of 5b

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

Step 1

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

Factor out $b^{2}$ .

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

Reorder $2$ and $b^{2}$ .

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

Add parentheses.

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

Step 2

Pull terms out from under the radical.

$| b | \sqrt{2 b} \sqrt[3]{5 b}$

Step 3

Multiply $| b | \sqrt{2 b} \sqrt[3]{5 b}$ .

Rewrite the expression using the least common index of $6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5 b}$ as ${(5 b)}^{\frac{1}{3}}$ .

$| b | ({(5 b)}^{\frac{1}{3}} \sqrt{2 b})$

Rewrite ${(5 b)}^{\frac{1}{3}}$ as ${(5 b)}^{\frac{2}{6}}$ .

$| b | ({(5 b)}^{\frac{2}{6}} \sqrt{2 b})$

Rewrite ${(5 b)}^{\frac{2}{6}}$ as $\sqrt[6]{{(5 b)}^{2}}$ .

$| b | (\sqrt[6]{{(5 b)}^{2}} \sqrt{2 b})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 b}$ as ${(2 b)}^{\frac{1}{2}}$ .

$| b | (\sqrt[6]{{(5 b)}^{2}} {(2 b)}^{\frac{1}{2}})$

Rewrite ${(2 b)}^{\frac{1}{2}}$ as ${(2 b)}^{\frac{3}{6}}$ .

$| b | (\sqrt[6]{{(5 b)}^{2}} {(2 b)}^{\frac{3}{6}})$

Rewrite ${(2 b)}^{\frac{3}{6}}$ as $\sqrt[6]{{(2 b)}^{3}}$ .

$| b | (\sqrt[6]{{(5 b)}^{2}} \sqrt[6]{{(2 b)}^{3}})$

Combine using the product rule for radicals.

$| b | \sqrt[6]{{(5 b)}^{2} {(2 b)}^{3}}$

Apply the product rule to $5 b$ .

$| b | \sqrt[6]{5^{2} b^{2} {(2 b)}^{3}}$

Apply the product rule to $2 b$ .

$| b | \sqrt[6]{5^{2} b^{2} \cdot 2^{3} b^{3}}$

Multiply $b^{2}$ by $b^{3}$ by adding the exponents.

Move $b^{3}$ .

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

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

$| b | \sqrt[6]{5^{2} b^{3 + 2} \cdot 2^{3}}$

Add $3$ and $2$ .

$| b | \sqrt[6]{5^{2} b^{5} \cdot 2^{3}}$

Raise $5$ to the power of $2$ .

$| b | \sqrt[6]{25 b^{5} \cdot 2^{3}}$

Raise $2$ to the power of $3$ .

$| b | \sqrt[6]{25 b^{5} \cdot 8}$

Multiply $8$ by $25$ .

$| b | \sqrt[6]{200 b^{5}}$

Simplify the Radical Expression square root of 2b^3 cube root of 5b

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