Algebra Examples

$\sqrt{z} \cdot \sqrt{30 z^{2}} \cdot \sqrt{35 z^{3}}$

Step 1

Reorder $30$ and $z^{2}$ .

$\sqrt{z} \cdot \sqrt{z^{2} \cdot 30} \cdot \sqrt{35 z^{3}}$

Step 2

Pull terms out from under the radical.

$\sqrt{z} \cdot (z \sqrt{30}) \cdot \sqrt{35 z^{3}}$

Step 3

Rewrite using the commutative property of multiplication.

$\sqrt{30} \sqrt{z} z \cdot \sqrt{35 z^{3}}$

Step 4

Combine using the product rule for radicals.

$\sqrt{30 z} z \cdot \sqrt{35 z^{3}}$

Step 5

Rewrite $35 z^{3}$ as $z^{2} \cdot (35 z)$ .

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Step 5.1

Factor out $z^{2}$ .

$\sqrt{30 z} z \cdot \sqrt{35 (z^{2} z)}$

Step 5.2

Reorder $35$ and $z^{2}$ .

$\sqrt{30 z} z \cdot \sqrt{z^{2} \cdot 35 z}$

Step 5.3

Add parentheses.

$\sqrt{30 z} z \cdot \sqrt{z^{2} \cdot (35 z)}$

Step 6

Pull terms out from under the radical.

$\sqrt{30 z} z \cdot (z \sqrt{35 z})$

Step 7

Multiply $z$ by $z$ by adding the exponents.

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Step 7.1

Move $z$ .

$\sqrt{30 z} (z \cdot z) \cdot \sqrt{35 z}$

Step 7.2

Multiply $z$ by $z$ .

$\sqrt{30 z} z^{2} \cdot \sqrt{35 z}$

Step 8

Multiply $\sqrt{30 z} z^{2} \sqrt{35 z}$ .

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Step 8.1

Combine using the product rule for radicals.

$z^{2} \sqrt{30 z (35 z)}$

Step 8.2

Multiply $35$ by $30$ .

$z^{2} \sqrt{1050 z \cdot z}$

Step 8.3

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

$z^{2} \sqrt{1050 (z^{1} z)}$

Step 8.4

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

$z^{2} \sqrt{1050 (z^{1} z^{1})}$

Step 8.5

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

$z^{2} \sqrt{1050 z^{1 + 1}}$

Step 8.6

Add $1$ and $1$ .

$z^{2} \sqrt{1050 z^{2}}$

Step 9

Rewrite $1050 z^{2}$ as ${(5 z)}^{2} \cdot 42$ .

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Step 9.1

Factor $25$ out of $1050$ .

$z^{2} \sqrt{25 (42) z^{2}}$

Step 9.2

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

$z^{2} \sqrt{5^{2} \cdot 42 z^{2}}$

Step 9.3

Move $42$ .

$z^{2} \sqrt{5^{2} z^{2} \cdot 42}$

Step 9.4

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

$z^{2} \sqrt{{(5 z)}^{2} \cdot 42}$

Step 10

Pull terms out from under the radical.

$z^{2} (5 z \sqrt{42})$

Step 11

Rewrite using the commutative property of multiplication.

$5 \sqrt{42} z^{2} z$

Step 12

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

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Step 12.1

Move $z$ .

$5 \sqrt{42} (z \cdot z^{2})$

Step 12.2

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

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Step 12.2.1

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

$5 \sqrt{42} (z^{1} z^{2})$

Step 12.2.2

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

$5 \sqrt{42} z^{1 + 2}$

Step 12.3

Add $1$ and $2$ .

$5 \sqrt{42} z^{3}$