Simplify (3 square root of 8a)*(8 square root of 3a)

$(3 \sqrt{8 a}) \cdot (8 \sqrt{3 a})$

Step 1

Rewrite $8 a$ as $2^{2} \cdot (2 a)$ .

Factor $4$ out of $8$ .

$3 \sqrt{4 (2) a} \cdot (8 \sqrt{3 a})$

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

$3 \sqrt{2^{2} \cdot 2 a} \cdot (8 \sqrt{3 a})$

Add parentheses.

$3 \sqrt{2^{2} \cdot (2 a)} \cdot (8 \sqrt{3 a})$

Step 2

Pull terms out from under the radical.

$3 (2 \sqrt{2 a}) \cdot (8 \sqrt{3 a})$

Step 3

Multiply $2$ by $3$ .

$6 \sqrt{2 a} \cdot (8 \sqrt{3 a})$

Step 4

Multiply $6 \sqrt{2 a} (8 \sqrt{3 a})$ .

Multiply $8$ by $6$ .

$48 \sqrt{2 a} \sqrt{3 a}$

Combine using the product rule for radicals.

$48 \sqrt{3 a (2 a)}$

Multiply $2$ by $3$ .

$48 \sqrt{6 a \cdot a}$

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

$48 \sqrt{6 (a^{1} a)}$

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

$48 \sqrt{6 (a^{1} a^{1})}$

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

$48 \sqrt{6 a^{1 + 1}}$

Add $1$ and $1$ .

$48 \sqrt{6 a^{2}}$

Step 5

Reorder $6$ and $a^{2}$ .

$48 \sqrt{a^{2} \cdot 6}$

Step 6

Pull terms out from under the radical.

$48 a \sqrt{6}$