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 $\sqrt{30 z} | z | (| z | \sqrt{35 z})$ .

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Step 7.1

Combine using the product rule for radicals.

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

Step 7.2

Multiply $30$ by $35$ .

$\sqrt{1050 z \cdot z} | z | | z |$

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

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

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

Step 7.6

Add $1$ and $1$ .

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

Step 7.7

To multiply absolute values, multiply the terms inside each absolute value.

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

Step 7.8

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

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

Step 7.9

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

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

Step 7.10

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

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

Step 7.11

Add $1$ and $1$ .

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

Step 8

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

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Step 8.1

Factor $25$ out of $1050$ .

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

Step 8.2

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

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

Step 8.3

Move $42$ .

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

Step 8.4

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

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

Step 9

Pull terms out from under the radical.

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

Step 10

Remove non-negative terms from the absolute value.

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

Step 11

Remove non-negative terms from the absolute value.

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