Algebra Examples

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

Step 1

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

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Step 1.1

Factor out $z^{2}$ .

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

Step 1.2

Move $y$ .

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

Step 1.3

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

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

Step 1.4

Add parentheses.

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

Step 2

Pull terms out from under the radical.

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

Step 3

Rewrite $x^{3} z^{5}$ as ${(x z^{2})}^{2} (x z)$ .

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Step 3.1

Factor out $x^{2}$ .

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

Step 3.2

Factor out $z^{4}$ .

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

Step 3.3

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

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

Step 3.4

Move $x$ .

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

Step 3.5

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

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

Step 3.6

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

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Step 5.1

Move $x$ .

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

Step 5.2

Multiply $x$ by $x$ .

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

Step 6

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

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Step 6.1

Move $z^{2}$ .

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

Step 6.2

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

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Step 6.2.1

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

$x^{2} (z^{2} z^{1}) \sqrt{y z} \cdot \sqrt{x z}$

Step 6.2.2

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

$x^{2} z^{2 + 1} \sqrt{y z} \cdot \sqrt{x z}$

Step 6.3

Add $2$ and $1$ .

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

Step 7

Multiply $x^{2} z^{3} \sqrt{y z} \sqrt{x z}$ .

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Step 7.1

Combine using the product rule for radicals.

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

Add $1$ and $1$ .

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

Step 8

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

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Step 8.1

Move $y$ .

$x^{2} z^{3} \sqrt{x z^{2} y}$

Step 8.2

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

$x^{2} z^{3} \sqrt{z^{2} x y}$

Step 8.3

Add parentheses.

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

Step 9

Pull terms out from under the radical.

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

Step 10

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

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Step 10.1

Move $z$ .

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

Step 10.2

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

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Step 10.2.1

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

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

Step 10.2.2

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

$x^{2} z^{1 + 3} \sqrt{x y}$

Step 10.3

Add $1$ and $3$ .

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