Simplify ( square root of 5x^2y)^3( square root of 8xy^3)^2

${(\sqrt{5 x^{2} y})}^{3} {(\sqrt{8 x y^{3}})}^{2}$

Step 1

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

Reorder $5$ and $x^{2}$ .

${\sqrt{x^{2} \cdot 5 y}}^{3} {(\sqrt{8 x y^{3}})}^{2}$

Add parentheses.

${\sqrt{x^{2} \cdot (5 y)}}^{3} {(\sqrt{8 x y^{3}})}^{2}$

Step 2

Pull terms out from under the radical.

${(x \sqrt{5 y})}^{3} {(\sqrt{8 x y^{3}})}^{2}$

Step 3

Apply the product rule to $x \sqrt{5 y}$ .

$x^{3} {\sqrt{5 y}}^{3} {(\sqrt{8 x y^{3}})}^{2}$

Step 4

Rewrite ${\sqrt{5 y}}^{3}$ as ${({(5 y)}^{3})}^{\frac{1}{2}}$ .

$x^{3} \sqrt{{(5 y)}^{3}} {(\sqrt{8 x y^{3}})}^{2}$

Step 5

Apply the product rule to $5 y$ .

$x^{3} \sqrt{5^{3} y^{3}} {(\sqrt{8 x y^{3}})}^{2}$

Step 6

Raise $5$ to the power of $3$ .

$x^{3} \sqrt{125 y^{3}} {(\sqrt{8 x y^{3}})}^{2}$

Step 7

Rewrite $125 y^{3}$ as ${(5 y)}^{2} \cdot (5 y)$ .

Factor $25$ out of $125$ .

$x^{3} \sqrt{25 (5) y^{3}} {(\sqrt{8 x y^{3}})}^{2}$

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

$x^{3} \sqrt{5^{2} \cdot 5 y^{3}} {(\sqrt{8 x y^{3}})}^{2}$

Factor out $y^{2}$ .

$x^{3} \sqrt{5^{2} \cdot 5 (y^{2} y)} {(\sqrt{8 x y^{3}})}^{2}$

Move $5$ .

$x^{3} \sqrt{5^{2} y^{2} \cdot 5 y} {(\sqrt{8 x y^{3}})}^{2}$

Rewrite $5^{2} y^{2}$ as ${(5 y)}^{2}$ .

$x^{3} \sqrt{{(5 y)}^{2} \cdot 5 y} {(\sqrt{8 x y^{3}})}^{2}$

Add parentheses.

$x^{3} \sqrt{{(5 y)}^{2} \cdot (5 y)} {(\sqrt{8 x y^{3}})}^{2}$

Step 8

Pull terms out from under the radical.

$x^{3} (5 y \sqrt{5 y}) {(\sqrt{8 x y^{3}})}^{2}$

Step 9

Rewrite using the commutative property of multiplication.

$5 x^{3} y \sqrt{5 y} {(\sqrt{8 x y^{3}})}^{2}$

Step 10

Rewrite $8 x y^{3}$ as ${(2 y)}^{2} \cdot (2 x y)$ .

Factor $4$ out of $8$ .

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

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

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

Factor out $y^{2}$ .

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

Move $x$ .

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

Move $2$ .

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

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

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

Add parentheses.

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

Add parentheses.

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

Step 11

Pull terms out from under the radical.

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

Step 12

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $2 y \sqrt{2 x y}$ .

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

Apply the product rule to $2 y$ .

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

Step 13

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

Move $y^{2}$ .

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

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

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

$5 x^{3} (y^{2} y^{1}) \sqrt{5 y} (2^{2} {\sqrt{2 x y}}^{2})$

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

$5 x^{3} y^{2 + 1} \sqrt{5 y} (2^{2} {\sqrt{2 x y}}^{2})$

Add $2$ and $1$ .

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

Step 14

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

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

Step 15

Rewrite ${\sqrt{2 x y}}^{2}$ as $2 x y$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x y}$ as ${(2 x y)}^{\frac{1}{2}}$ .

$5 x^{3} y^{3} \sqrt{5 y} (4 {({(2 x y)}^{\frac{1}{2}})}^{2})$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$5 x^{3} y^{3} \sqrt{5 y} (4 {(2 x y)}^{\frac{1}{2} \cdot 2})$

Combine $\frac{1}{2}$ and $2$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$5 x^{3} y^{3} \sqrt{5 y} (4 {(2 x y)}^{1})$

Simplify.

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

Step 16

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

Move $x$ .

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

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

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

$5 (x^{1} x^{3}) y^{3} \sqrt{5 y} (4 (2 y))$

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

$5 x^{1 + 3} y^{3} \sqrt{5 y} (4 (2 y))$

Add $1$ and $3$ .

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

Step 17

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

Move $y$ .

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

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

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

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

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

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

Add $1$ and $3$ .

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

Step 18

Multiply $4$ by $2$ .

$5 x^{4} y^{4} \sqrt{5 y} \cdot 8$

Step 19

Multiply $8$ by $5$ .

$40 x^{4} y^{4} \sqrt{5 y}$

Simplify ( square root of 5x^2y)^3( square root of 8xy^3)^2

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