Simplify (6 square root of 22a^2)(4 square root of 2a^6)

$(6 \sqrt{22 a^{2}}) (4 \sqrt{2 a^{6}})$

Reorder $22$ and $a^{2}$ .

$6 \sqrt{a^{2} \cdot 22} (4 \sqrt{2 a^{6}})$

Pull terms out from under the radical.

$6 (a \sqrt{22}) (4 \sqrt{2 a^{6}})$

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

Rewrite $a^{6}$ as ${(a^{3})}^{2}$ .

$6 a \sqrt{22} (4 \sqrt{2 {(a^{3})}^{2}})$

Reorder $2$ and ${(a^{3})}^{2}$ .

$6 a \sqrt{22} (4 \sqrt{{(a^{3})}^{2} \cdot 2})$

$6 a \sqrt{22} (4 \sqrt{{(a^{3})}^{2} \cdot 2})$

Pull terms out from under the radical.

$6 a \sqrt{22} (4 (a^{3} \sqrt{2}))$

Multiply $a$ by $a^{3}$ by adding the exponents.

Move $a^{3}$ .

$6 (a^{3} a) \sqrt{22} (4 (\sqrt{2}))$

Multiply $a^{3}$ by $a$ .

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

$6 (a^{3} a^{1}) \sqrt{22} (4 (\sqrt{2}))$

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

$6 a^{3 + 1} \sqrt{22} (4 (\sqrt{2}))$

$6 a^{3 + 1} \sqrt{22} (4 (\sqrt{2}))$

Add $3$ and $1$ .

$6 a^{4} \sqrt{22} (4 (\sqrt{2}))$

$6 a^{4} \sqrt{22} (4 (\sqrt{2}))$

Multiply $6 a^{4} \sqrt{22} (4 \sqrt{2})$ .

Multiply $4$ by $6$ .

$24 a^{4} \sqrt{22} \sqrt{2}$

Combine using the product rule for radicals.

$24 a^{4} \sqrt{2 \cdot 22}$

Multiply $2$ by $22$ .

$24 a^{4} \sqrt{44}$

$24 a^{4} \sqrt{44}$

Rewrite $44$ as $2^{2} \cdot 11$ .

Factor $4$ out of $44$ .

$24 a^{4} \sqrt{4 (11)}$

Rewrite $4$ as $2^{2}$ .

$24 a^{4} \sqrt{2^{2} \cdot 11}$

$24 a^{4} \sqrt{2^{2} \cdot 11}$

Pull terms out from under the radical.

$24 a^{4} (2 \sqrt{11})$

Multiply $2$ by $24$ .

$48 a^{4} \sqrt{11}$

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