Simplify (3 square root of 32y)(4 square root of 25y)

$(3 \sqrt{32 y}) (4 \sqrt{25 y})$

Step 1

Rewrite $32 y$ as ${(2^{2})}^{2} \cdot (2 y)$ .

Factor $16$ out of $32$ .

$3 \sqrt{16 (2) y} (4 \sqrt{25 y})$

Rewrite $16$ as $4^{2}$ .

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

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

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

Add parentheses.

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

Step 2

Pull terms out from under the radical.

$3 (2^{2} \sqrt{2 y}) (4 \sqrt{25 y})$

Step 3

Simplify the expression.

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

$3 (4 \sqrt{2 y}) (4 \sqrt{25 y})$

Multiply $4$ by $3$ .

$12 \sqrt{2 y} (4 \sqrt{25 y})$

Rewrite $25$ as $5^{2}$ .

$12 \sqrt{2 y} (4 \sqrt{5^{2} y})$

Step 4

Pull terms out from under the radical.

$12 \sqrt{2 y} (4 (5 \sqrt{y}))$

Step 5

Multiply $5$ by $4$ .

$12 \sqrt{2 y} (20 \sqrt{y})$

Step 6

Multiply $12 \sqrt{2 y} (20 \sqrt{y})$ .

Multiply $20$ by $12$ .

$240 \sqrt{2 y} \sqrt{y}$

Combine using the product rule for radicals.

$240 \sqrt{y (2 y)}$

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

$240 \sqrt{2 (y^{1} y)}$

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

$240 \sqrt{2 (y^{1} y^{1})}$

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

$240 \sqrt{2 y^{1 + 1}}$

Add $1$ and $1$ .

$240 \sqrt{2 y^{2}}$

Step 7

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

$240 \sqrt{y^{2} \cdot 2}$

Step 8

Pull terms out from under the radical.

$240 y \sqrt{2}$

Simplify (3 square root of 32y)(4 square root of 25y)

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