Find the Product square root of 5x^8y^2* square root of 10x^3* square root of 12y

$\sqrt{5 x^{8} y^{2}} \cdot \sqrt{10 x^{3}} \cdot \sqrt{12 y}$

Step 1

Rewrite $5 x^{8} y^{2}$ as ${(x^{4} y)}^{2} \cdot 5$ .

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

$\sqrt{5 {(x^{4})}^{2} y^{2}} \cdot \sqrt{10 x^{3}} \cdot \sqrt{12 y}$

Move $5$ .

$\sqrt{{(x^{4})}^{2} y^{2} \cdot 5} \cdot \sqrt{10 x^{3}} \cdot \sqrt{12 y}$

Rewrite ${(x^{4})}^{2} y^{2}$ as ${(x^{4} y)}^{2}$ .

$\sqrt{{(x^{4} y)}^{2} \cdot 5} \cdot \sqrt{10 x^{3}} \cdot \sqrt{12 y}$

Step 2

Pull terms out from under the radical.

$x^{4} y \sqrt{5} \cdot \sqrt{10 x^{3}} \cdot \sqrt{12 y}$

Step 3

Rewrite $10 x^{3}$ as $x^{2} \cdot (10 x)$ .

Factor out $x^{2}$ .

$x^{4} y \sqrt{5} \cdot \sqrt{10 (x^{2} x)} \cdot \sqrt{12 y}$

Reorder $10$ and $x^{2}$ .

$x^{4} y \sqrt{5} \cdot \sqrt{x^{2} \cdot 10 x} \cdot \sqrt{12 y}$

Add parentheses.

$x^{4} y \sqrt{5} \cdot \sqrt{x^{2} \cdot (10 x)} \cdot \sqrt{12 y}$

Step 4

Pull terms out from under the radical.

$x^{4} y \sqrt{5} \cdot (x \sqrt{10 x}) \cdot \sqrt{12 y}$

Step 5

Multiply $x^{4}$ by $x$ by adding the exponents.

Move $x$ .

$x \cdot x^{4} y \sqrt{5} \cdot \sqrt{10 x} \cdot \sqrt{12 y}$

Multiply $x$ by $x^{4}$ .

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

$x^{1} x^{4} y \sqrt{5} \cdot \sqrt{10 x} \cdot \sqrt{12 y}$

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

$x^{1 + 4} y \sqrt{5} \cdot \sqrt{10 x} \cdot \sqrt{12 y}$

Add $1$ and $4$ .

$x^{5} y \sqrt{5} \cdot \sqrt{10 x} \cdot \sqrt{12 y}$

Step 6

Multiply $x^{5} y \sqrt{5} \sqrt{10 x}$ .

Combine using the product rule for radicals.

$x^{5} y \sqrt{10 x \cdot 5} \cdot \sqrt{12 y}$

Multiply $5$ by $10$ .

$x^{5} y \sqrt{50 x} \cdot \sqrt{12 y}$

Step 7

Rewrite $50 x$ as $5^{2} \cdot (2 x)$ .

Factor $25$ out of $50$ .

$x^{5} y \sqrt{25 (2) x} \cdot \sqrt{12 y}$

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

$x^{5} y \sqrt{5^{2} \cdot 2 x} \cdot \sqrt{12 y}$

Add parentheses.

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

Step 8

Pull terms out from under the radical.

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

Step 9

Rewrite using the commutative property of multiplication.

$5 x^{5} y \sqrt{2 x} \cdot \sqrt{12 y}$

Step 10

Rewrite $12 y$ as $2^{2} \cdot (3 y)$ .

Factor $4$ out of $12$ .

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

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

$5 x^{5} y \sqrt{2 x} \cdot \sqrt{2^{2} \cdot 3 y}$

Add parentheses.

$5 x^{5} y \sqrt{2 x} \cdot \sqrt{2^{2} \cdot (3 y)}$

Step 11

Pull terms out from under the radical.

$5 x^{5} y \sqrt{2 x} \cdot (2 \sqrt{3 y})$

Step 12

Multiply $5 x^{5} y \sqrt{2 x} (2 \sqrt{3 y})$ .

Multiply $2$ by $5$ .

$10 x^{5} y \sqrt{2 x} \sqrt{3 y}$

Combine using the product rule for radicals.

$10 x^{5} y \sqrt{3 y (2 x)}$

Multiply $2$ by $3$ .

$10 x^{5} y \sqrt{6 y x}$

Find the Product square root of 5x^8y^2* square root of 10x^3* square root of 12y

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