Simplify fifth root of 72i^4b^3 fifth root of 216i^8b^11

$\sqrt[5]{72 i^{4} b^{3}} \sqrt[5]{216 i^{8} b^{11}}$

Step 1

Rewrite $i^{4}$ as $1$ .

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$\sqrt[5]{72 {(i^{2})}^{2} b^{3}} \sqrt[5]{216 i^{8} b^{11}}$

Rewrite $i^{2}$ as $- 1$ .

$\sqrt[5]{72 {(- 1)}^{2} b^{3}} \sqrt[5]{216 i^{8} b^{11}}$

Raise $- 1$ to the power of $2$ .

$\sqrt[5]{72 \cdot 1 b^{3}} \sqrt[5]{216 i^{8} b^{11}}$

Step 2

Multiply $72$ by $1$ .

$\sqrt[5]{72 b^{3}} \sqrt[5]{216 i^{8} b^{11}}$

Step 3

Rewrite $i^{8}$ as ${(i^{4})}^{2}$ .

$\sqrt[5]{72 b^{3}} \sqrt[5]{216 {(i^{4})}^{2} b^{11}}$

Step 4

Rewrite $i^{4}$ as $1$ .

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$\sqrt[5]{72 b^{3}} \sqrt[5]{216 {({(i^{2})}^{2})}^{2} b^{11}}$

Rewrite $i^{2}$ as $- 1$ .

$\sqrt[5]{72 b^{3}} \sqrt[5]{216 {({(- 1)}^{2})}^{2} b^{11}}$

Raise $- 1$ to the power of $2$ .

$\sqrt[5]{72 b^{3}} \sqrt[5]{216 \cdot 1^{2} b^{11}}$

Step 5

One to any power is one.

$\sqrt[5]{72 b^{3}} \sqrt[5]{216 \cdot 1 b^{11}}$

Step 6

Multiply $216$ by $1$ .

$\sqrt[5]{72 b^{3}} \sqrt[5]{216 b^{11}}$

Step 7

Rewrite $216 b^{11}$ as ${(b^{2})}^{5} \cdot (216 b)$ .

Factor out $b^{10}$ .

$\sqrt[5]{72 b^{3}} \sqrt[5]{216 (b^{10} b)}$

Rewrite $b^{10}$ as ${(b^{2})}^{5}$ .

$\sqrt[5]{72 b^{3}} \sqrt[5]{216 ({(b^{2})}^{5} b)}$

Reorder $216$ and ${(b^{2})}^{5}$ .

$\sqrt[5]{72 b^{3}} \sqrt[5]{{(b^{2})}^{5} \cdot 216 b}$

Add parentheses.

$\sqrt[5]{72 b^{3}} \sqrt[5]{{(b^{2})}^{5} \cdot (216 b)}$

Step 8

Pull terms out from under the radical.

$\sqrt[5]{72 b^{3}} (b^{2} \sqrt[5]{216 b})$

Step 9

Multiply $\sqrt[5]{72 b^{3}} (b^{2} \sqrt[5]{216 b})$ .

Combine using the product rule for radicals.

$b^{2} \sqrt[5]{72 b^{3} (216 b)}$

Multiply $216$ by $72$ .

$b^{2} \sqrt[5]{15552 b^{3} b}$

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

$b^{2} \sqrt[5]{15552 (b^{1} b^{3})}$

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

$b^{2} \sqrt[5]{15552 b^{1 + 3}}$

Add $1$ and $3$ .

$b^{2} \sqrt[5]{15552 b^{4}}$

Step 10

Rewrite $15552 b^{4}$ as $6^{5} \cdot (2 b^{4})$ .

Factor $7776$ out of $15552$ .

$b^{2} \sqrt[5]{7776 (2) b^{4}}$

Rewrite $7776$ as $6^{5}$ .

$b^{2} \sqrt[5]{6^{5} \cdot 2 b^{4}}$

Add parentheses.

$b^{2} \sqrt[5]{6^{5} \cdot (2 b^{4})}$

Step 11

Pull terms out from under the radical.

$b^{2} (6 \sqrt[5]{2 b^{4}})$

Step 12

Rewrite using the commutative property of multiplication.

$6 b^{2} \sqrt[5]{2 b^{4}}$

Simplify fifth root of 72i^4b^3 fifth root of 216i^8b^11

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