Algebra Examples

$\sqrt[4]{16 x^{2} y^{6}} \cdot \sqrt{28 x^{10} y^{22}}$

Step 1

Rewrite $16 x^{2} y^{6}$ as ${(2 y)}^{4} (x^{2} y^{2})$ .

Tap for more steps...

Step 1.1

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

$\sqrt[4]{2^{4} x^{2} y^{6}} \cdot \sqrt{28 x^{10} y^{22}}$

Step 1.2

Factor out $y^{4}$ .

$\sqrt[4]{2^{4} x^{2} (y^{4} y^{2})} \cdot \sqrt{28 x^{10} y^{22}}$

Step 1.3

Move $x^{2}$ .

$\sqrt[4]{2^{4} y^{4} x^{2} y^{2}} \cdot \sqrt{28 x^{10} y^{22}}$

Step 1.4

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

$\sqrt[4]{{(2 y)}^{4} x^{2} y^{2}} \cdot \sqrt{28 x^{10} y^{22}}$

Step 1.5

Add parentheses.

$\sqrt[4]{{(2 y)}^{4} (x^{2} y^{2})} \cdot \sqrt{28 x^{10} y^{22}}$

Step 2

Pull terms out from under the radical.

$2 y \sqrt[4]{x^{2} y^{2}} \cdot \sqrt{28 x^{10} y^{22}}$

Step 3

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

$2 y \sqrt[4]{{(x y)}^{2}} \cdot \sqrt{28 x^{10} y^{22}}$

Step 4

Rewrite $\sqrt[4]{{(x y)}^{2}}$ as $\sqrt{\sqrt{{(x y)}^{2}}}$ .

$2 y \sqrt{\sqrt{{(x y)}^{2}}} \cdot \sqrt{28 x^{10} y^{22}}$

Step 5

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

$2 y \sqrt{x y} \cdot \sqrt{28 x^{10} y^{22}}$

Step 6

Rewrite $28 x^{10} y^{22}$ as ${(2 x^{5} y^{11})}^{2} \cdot 7$ .

Tap for more steps...

Step 6.1

Factor $4$ out of $28$ .

$2 y \sqrt{x y} \cdot \sqrt{4 (7) x^{10} y^{22}}$

Step 6.2

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

$2 y \sqrt{x y} \cdot \sqrt{2^{2} \cdot 7 x^{10} y^{22}}$

Step 6.3

Rewrite $x^{10}$ as ${(x^{5})}^{2}$ .

$2 y \sqrt{x y} \cdot \sqrt{2^{2} \cdot 7 {(x^{5})}^{2} y^{22}}$

Step 6.4

Rewrite $y^{22}$ as ${(y^{11})}^{2}$ .

$2 y \sqrt{x y} \cdot \sqrt{2^{2} \cdot 7 {(x^{5})}^{2} {(y^{11})}^{2}}$

Step 6.5

Move $7$ .

$2 y \sqrt{x y} \cdot \sqrt{2^{2} {(x^{5})}^{2} {(y^{11})}^{2} \cdot 7}$

Step 6.6

Rewrite $2^{2} {(x^{5})}^{2} {(y^{11})}^{2}$ as ${(2 x^{5} y^{11})}^{2}$ .

$2 y \sqrt{x y} \cdot \sqrt{{(2 x^{5} y^{11})}^{2} \cdot 7}$

Step 7

Pull terms out from under the radical.

$2 y \sqrt{x y} \cdot (2 x^{5} y^{11} \sqrt{7})$

Step 8

Multiply $y$ by $y^{11}$ by adding the exponents.

Tap for more steps...

Step 8.1

Move $y^{11}$ .

$2 (y^{11} y) \sqrt{x y} \cdot (2 x^{5} \sqrt{7})$

Step 8.2

Multiply $y^{11}$ by $y$ .

Tap for more steps...

Step 8.2.1

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

$2 (y^{11} y^{1}) \sqrt{x y} \cdot (2 x^{5} \sqrt{7})$

Step 8.2.2

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

$2 y^{11 + 1} \sqrt{x y} \cdot (2 x^{5} \sqrt{7})$

Step 8.3

Add $11$ and $1$ .

$2 y^{12} \sqrt{x y} \cdot (2 x^{5} \sqrt{7})$

Step 9

Multiply $2 y^{12} \sqrt{x y} (2 x^{5} \sqrt{7})$ .

Tap for more steps...

Step 9.1

Multiply $2$ by $2$ .

$4 y^{12} \sqrt{x y} (x^{5} \sqrt{7})$

Step 9.2

Combine using the product rule for radicals.

$4 y^{12} (x^{5} \sqrt{x y \cdot 7})$

Step 10

Reorder factors in $4 y^{12} (x^{5} \sqrt{x y \cdot 7})$ .

$4 y^{12} x^{5} \sqrt{7 x y}$