Algebra Examples

$5 \sqrt{14 z^{2}} \cdot 4 \sqrt{21 z^{3}}$

Step 1

Reorder $14$ and $z^{2}$ .

$5 \sqrt{z^{2} \cdot 14} \cdot 4 \sqrt{21 z^{3}}$

Step 2

Pull terms out from under the radical.

$5 (| z | \sqrt{14}) \cdot 4 \sqrt{21 z^{3}}$

Step 3

Multiply $4$ by $5$ .

$20 | z | \sqrt{14} \sqrt{21 z^{3}}$

Step 4

Rewrite $21 z^{3}$ as $z^{2} \cdot (21 z)$ .

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Step 4.1

Factor out $z^{2}$ .

$20 | z | \sqrt{14} \sqrt{21 (z^{2} z)}$

Step 4.2

Reorder $21$ and $z^{2}$ .

$20 | z | \sqrt{14} \sqrt{z^{2} \cdot 21 z}$

Step 4.3

Add parentheses.

$20 | z | \sqrt{14} \sqrt{z^{2} \cdot (21 z)}$

Step 5

Pull terms out from under the radical.

$20 | z | \sqrt{14} (| z | \sqrt{21 z})$

Step 6

Multiply $20 | z | \sqrt{14} (| z | \sqrt{21 z})$ .

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Step 6.1

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

$20 | z \cdot z | \sqrt{14} \sqrt{21 z}$

Step 6.2

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

$20 | z^{1} z | \sqrt{14} \sqrt{21 z}$

Step 6.3

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

$20 | z^{1} z^{1} | \sqrt{14} \sqrt{21 z}$

Step 6.4

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

$20 | z^{1 + 1} | \sqrt{14} \sqrt{21 z}$

Step 6.5

Add $1$ and $1$ .

$20 | z^{2} | \sqrt{14} \sqrt{21 z}$

Step 6.6

Combine using the product rule for radicals.

$20 | z^{2} | \sqrt{21 z \cdot 14}$

Step 6.7

Multiply $14$ by $21$ .

$20 | z^{2} | \sqrt{294 z}$

Step 7

Remove non-negative terms from the absolute value.

$20 z^{2} \sqrt{294 z}$

Step 8

Rewrite $294 z$ as $7^{2} \cdot (6 z)$ .

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Step 8.1

Factor $49$ out of $294$ .

$20 z^{2} \sqrt{49 (6) z}$

Step 8.2

Rewrite $49$ as $7^{2}$ .

$20 z^{2} \sqrt{7^{2} \cdot 6 z}$

Step 8.3

Add parentheses.

$20 z^{2} \sqrt{7^{2} \cdot (6 z)}$

Step 9

Pull terms out from under the radical.

$20 z^{2} (| 7 | \sqrt{6 z})$

Step 10

The absolute value is the distance between a number and zero. The distance between $0$ and $7$ is $7$ .

$20 z^{2} (7 \sqrt{6 z})$

Step 11

Multiply $7$ by $20$ .

$140 z^{2} \sqrt{6 z}$