Simplify (-2 square root of 7z^3)(3 square root of 14z^8)

$(- 2 \sqrt{7 z^{3}}) (3 \sqrt{14 z^{8}})$

Step 1

Rewrite $7 z^{3}$ as $z^{2} \cdot (7 z)$ .

Factor out $z^{2}$ .

$- 2 \sqrt{7 (z^{2} z)} (3 \sqrt{14 z^{8}})$

Reorder $7$ and $z^{2}$ .

$- 2 \sqrt{z^{2} \cdot 7 z} (3 \sqrt{14 z^{8}})$

Add parentheses.

$- 2 \sqrt{z^{2} \cdot (7 z)} (3 \sqrt{14 z^{8}})$

Step 2

Pull terms out from under the radical.

$- 2 (z \sqrt{7 z}) (3 \sqrt{14 z^{8}})$

Step 3

Rewrite $14 z^{8}$ as ${({(z^{2})}^{2})}^{2} \cdot 14$ .

Rewrite $z^{8}$ as ${(z^{4})}^{2}$ .

$- 2 z \sqrt{7 z} (3 \sqrt{14 {(z^{4})}^{2}})$

Reorder $14$ and ${(z^{4})}^{2}$ .

$- 2 z \sqrt{7 z} (3 \sqrt{{(z^{4})}^{2} \cdot 14})$

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

$- 2 z \sqrt{7 z} (3 \sqrt{{({(z^{2})}^{2})}^{2} \cdot 14})$

Step 4

Pull terms out from under the radical.

$- 2 z \sqrt{7 z} (3 ({(z^{2})}^{2} \sqrt{14}))$

Step 5

Multiply the exponents in ${(z^{2})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 2 z \sqrt{7 z} (3 (z^{2 \cdot 2} \sqrt{14}))$

Multiply $2$ by $2$ .

$- 2 z \sqrt{7 z} (3 (z^{4} \sqrt{14}))$

Step 6

Multiply $z$ by $z^{4}$ by adding the exponents.

Move $z^{4}$ .

$- 2 (z^{4} z) \sqrt{7 z} (3 (\sqrt{14}))$

Multiply $z^{4}$ by $z$ .

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

$- 2 (z^{4} z^{1}) \sqrt{7 z} (3 (\sqrt{14}))$

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

$- 2 z^{4 + 1} \sqrt{7 z} (3 (\sqrt{14}))$

Add $4$ and $1$ .

$- 2 z^{5} \sqrt{7 z} (3 (\sqrt{14}))$

Step 7

Multiply $- 2 z^{5} \sqrt{7 z} (3 \sqrt{14})$ .

Multiply $3$ by $- 2$ .

$- 6 z^{5} \sqrt{7 z} \sqrt{14}$

Combine using the product rule for radicals.

$- 6 z^{5} \sqrt{14 (7 z)}$

Multiply $14$ by $7$ .

$- 6 z^{5} \sqrt{98 z}$

Step 8

Rewrite $98 z$ as $7^{2} \cdot (2 z)$ .

Factor $49$ out of $98$ .

$- 6 z^{5} \sqrt{49 (2) z}$

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

$- 6 z^{5} \sqrt{7^{2} \cdot 2 z}$

Add parentheses.

$- 6 z^{5} \sqrt{7^{2} \cdot (2 z)}$

Step 9

Pull terms out from under the radical.

$- 6 z^{5} (7 \sqrt{2 z})$

Step 10

Multiply $7$ by $- 6$ .

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

Simplify (-2 square root of 7z^3)(3 square root of 14z^8)

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