Simplify 5 square root of 3a^3*4 square root of 25a^4

$5 \sqrt{3 a^{3}} \cdot 4 \sqrt{25 a^{4}}$

Step 1

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

Factor out $a^{2}$ .

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

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

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

Add parentheses.

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

Step 2

Pull terms out from under the radical.

$5 (a \sqrt{3 a}) \cdot (4 \sqrt{25 a^{4}})$

Step 3

Rewrite $25 a^{4}$ as ${(5 a^{2})}^{2}$ .

$5 a \sqrt{3 a} \cdot (4 \sqrt{{(5 a^{2})}^{2}})$

Step 4

Pull terms out from under the radical, assuming positive real numbers.

$5 a \sqrt{3 a} \cdot (4 (5 a^{2}))$

Step 5

Multiply $a$ by $a^{2}$ by adding the exponents.

Move $a^{2}$ .

$5 (a^{2} a) \sqrt{3 a} \cdot (4 \cdot (5))$

Multiply $a^{2}$ by $a$ .

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

$5 (a^{2} a^{1}) \sqrt{3 a} \cdot (4 \cdot (5))$

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

$5 a^{2 + 1} \sqrt{3 a} \cdot (4 \cdot (5))$

Add $2$ and $1$ .

$5 a^{3} \sqrt{3 a} \cdot (4 \cdot (5))$

Step 6

Multiply $4$ by $5$ .

$5 a^{3} \sqrt{3 a} \cdot 20$

Step 7

Multiply $20$ by $5$ .

$100 a^{3} \sqrt{3 a}$