Algebra Examples

$3 x \sqrt[4]{8 x^{2} y^{2}} \cdot 5 y \sqrt[4]{8 x^{2} y^{4}}$

Step 1

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

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Step 1.1

Move $x^{2}$ .

$3 x \sqrt[4]{8 x^{2} y^{2}} \cdot (5 y \sqrt[4]{8 y^{4} x^{2}})$

Step 1.2

Reorder $8$ and $y^{4}$ .

$3 x \sqrt[4]{8 x^{2} y^{2}} \cdot (5 y \sqrt[4]{y^{4} \cdot 8 x^{2}})$

Step 1.3

Add parentheses.

$3 x \sqrt[4]{8 x^{2} y^{2}} \cdot (5 y \sqrt[4]{y^{4} \cdot (8 x^{2})})$

Step 2

Pull terms out from under the radical.

$3 x \sqrt[4]{8 x^{2} y^{2}} \cdot (5 y (y \sqrt[4]{8 x^{2}}))$

Step 3

Multiply $y$ by $y$ by adding the exponents.

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Step 3.1

Move $y$ .

$3 x \sqrt[4]{8 x^{2} y^{2}} \cdot (5 (y \cdot y) \sqrt[4]{8 x^{2}})$

Step 3.2

Multiply $y$ by $y$ .

$3 x \sqrt[4]{8 x^{2} y^{2}} \cdot (5 y^{2} \sqrt[4]{8 x^{2}})$

Step 4

Multiply $3 x \sqrt[4]{8 x^{2} y^{2}} (5 y^{2} \sqrt[4]{8 x^{2}})$ .

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Step 4.1

Multiply $5$ by $3$ .

$15 x \sqrt[4]{8 x^{2} y^{2}} (y^{2} \sqrt[4]{8 x^{2}})$

Step 4.2

Combine using the product rule for radicals.

$15 x (y^{2} \sqrt[4]{8 x^{2} y^{2} (8 x^{2})})$

Step 4.3

Multiply $8$ by $8$ .

$15 x (y^{2} \sqrt[4]{64 x^{2} y^{2} x^{2}})$

Step 4.4

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

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Step 4.4.1

Move $x^{2}$ .

$15 x (y^{2} \sqrt[4]{64 (x^{2} x^{2}) y^{2}})$

Step 4.4.2

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

$15 x (y^{2} \sqrt[4]{64 x^{2 + 2} y^{2}})$

Step 4.4.3

Add $2$ and $2$ .

$15 x (y^{2} \sqrt[4]{64 x^{4} y^{2}})$

Step 5

Rewrite $64 x^{4} y^{2}$ as ${(2 x)}^{4} \cdot (4 y^{2})$ .

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Step 5.1

Factor $16$ out of $64$ .

$15 x (y^{2} \sqrt[4]{16 (4) x^{4} y^{2}})$

Step 5.2

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

$15 x (y^{2} \sqrt[4]{2^{4} \cdot 4 x^{4} y^{2}})$

Step 5.3

Move $4$ .

$15 x (y^{2} \sqrt[4]{2^{4} x^{4} \cdot 4 y^{2}})$

Step 5.4

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

$15 x (y^{2} \sqrt[4]{{(2 x)}^{4} \cdot 4 y^{2}})$

Step 5.5

Add parentheses.

$15 x (y^{2} \sqrt[4]{{(2 x)}^{4} \cdot (4 y^{2})})$

Step 6

Pull terms out from under the radical.

$15 x (y^{2} (2 x \sqrt[4]{4 y^{2}}))$

Step 7

Multiply $x$ by $x$ by adding the exponents.

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Step 7.1

Move $x$ .

$15 (x \cdot x) (y^{2} (2 \sqrt[4]{4 y^{2}}))$

Step 7.2

Multiply $x$ by $x$ .

$15 x^{2} (y^{2} (2 \sqrt[4]{4 y^{2}}))$

Step 8

Rewrite using the commutative property of multiplication.

$15 x^{2} (2 y^{2} \sqrt[4]{4 y^{2}})$

Step 9

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

$15 x^{2} (2 y^{2} \sqrt[4]{{(2 y)}^{2}})$

Step 10

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

$15 x^{2} (2 y^{2} \sqrt{\sqrt{{(2 y)}^{2}}})$

Step 11

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

$15 x^{2} (2 y^{2} \sqrt{2 y})$

Step 12

Multiply $2$ by $15$ .

$30 x^{2} y^{2} \sqrt{2 y}$