Algebra Examples

$\sqrt{2 x^{3} y z^{4}} \cdot \sqrt{32 x^{2} y^{2} z^{4}}$

Step 1

Rewrite $2 x^{3} y z^{4}$ as ${(x z^{2})}^{2} \cdot (2 x y)$ .

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Step 1.1

Factor out $x^{2}$ .

$\sqrt{2 (x^{2} x) y z^{4}} \cdot \sqrt{32 x^{2} y^{2} z^{4}}$

Step 1.2

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

$\sqrt{2 (x^{2} x) y {(z^{2})}^{2}} \cdot \sqrt{32 x^{2} y^{2} z^{4}}$

Step 1.3

Move $y$ .

$\sqrt{2 (x^{2} x) {(z^{2})}^{2} y} \cdot \sqrt{32 x^{2} y^{2} z^{4}}$

Step 1.4

Move $x$ .

$\sqrt{2 (x^{2}) {(z^{2})}^{2} x y} \cdot \sqrt{32 x^{2} y^{2} z^{4}}$

Step 1.5

Move $2$ .

$\sqrt{x^{2} {(z^{2})}^{2} \cdot 2 x y} \cdot \sqrt{32 x^{2} y^{2} z^{4}}$

Step 1.6

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

$\sqrt{{(x z^{2})}^{2} \cdot 2 x y} \cdot \sqrt{32 x^{2} y^{2} z^{4}}$

Step 1.7

Add parentheses.

$\sqrt{{(x z^{2})}^{2} \cdot 2 (x y)} \cdot \sqrt{32 x^{2} y^{2} z^{4}}$

Step 1.8

Add parentheses.

$\sqrt{{(x z^{2})}^{2} \cdot (2 x y)} \cdot \sqrt{32 x^{2} y^{2} z^{4}}$

Step 2

Pull terms out from under the radical.

$x z^{2} \sqrt{2 x y} \cdot \sqrt{32 x^{2} y^{2} z^{4}}$

Step 3

Rewrite $32 x^{2} y^{2} z^{4}$ as ${(4 x y z^{2})}^{2} \cdot 2$ .

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Step 3.1

Factor $16$ out of $32$ .

$x z^{2} \sqrt{2 x y} \cdot \sqrt{16 (2) x^{2} y^{2} z^{4}}$

Step 3.2

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

$x z^{2} \sqrt{2 x y} \cdot \sqrt{4^{2} \cdot 2 x^{2} y^{2} z^{4}}$

Step 3.3

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

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

Step 3.4

Move $2$ .

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

Step 3.5

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

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

Step 4

Pull terms out from under the radical.

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

Step 5

Rewrite using the commutative property of multiplication.

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

Step 6

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

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Step 6.1

Move $x$ .

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

Step 6.2

Multiply $x$ by $x$ .

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

Step 7

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

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Step 7.1

Move $z^{2}$ .

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

Step 7.2

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

$4 \sqrt{2} x^{2} z^{2 + 2} \sqrt{2 x y} y$

Step 7.3

Add $2$ and $2$ .

$4 \sqrt{2} x^{2} z^{4} \sqrt{2 x y} y$

Step 8

Multiply $4 \sqrt{2} x^{2} z^{4} \sqrt{2 x y}$ .

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Step 8.1

Combine using the product rule for radicals.

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

Step 8.2

Multiply $2$ by $2$ .

$4 x^{2} z^{4} \sqrt{4 x y} y$

Step 9

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

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Step 9.1

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

$4 x^{2} z^{4} \sqrt{2^{2} x y} y$

Step 9.2

Add parentheses.

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

Step 10

Pull terms out from under the radical.

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

Step 11

Multiply $2$ by $4$ .

$8 x^{2} z^{4} \sqrt{x y} y$