Simplify (8 square root of -84)(5 square root of -7)

$(8 \sqrt{- 84}) (5 \sqrt{- 7})$

Step 1

Rewrite $- 84$ as $- 1 (84)$ .

$8 \sqrt{- 1 (84)} (5 \sqrt{- 7})$

Step 2

Rewrite $\sqrt{- 1 (84)}$ as $\sqrt{- 1} \cdot \sqrt{84}$ .

$8 (\sqrt{- 1} \cdot \sqrt{84}) (5 \sqrt{- 7})$

Step 3

Rewrite $\sqrt{- 1}$ as $i$ .

$8 (i \cdot \sqrt{84}) (5 \sqrt{- 7})$

Step 4

Rewrite $- 7$ as $- 1 (7)$ .

$8 (i \cdot \sqrt{84}) (5 \sqrt{- 1 (7)})$

Step 5

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

$8 (i \cdot \sqrt{84}) (5 (\sqrt{- 1} \cdot \sqrt{7}))$

Step 6

Rewrite $\sqrt{- 1}$ as $i$ .

$8 (i \cdot \sqrt{84}) (5 (i \cdot \sqrt{7}))$

Step 7

Rewrite $84$ as $2^{2} \cdot 21$ .

Factor $4$ out of $84$ .

$8 (i \cdot \sqrt{4 (21)}) (5 (i \cdot \sqrt{7}))$

Rewrite $4$ as $2^{2}$ .

$8 (i \cdot \sqrt{2^{2} \cdot 21}) (5 (i \cdot \sqrt{7}))$

Step 8

Pull terms out from under the radical.

$8 (i \cdot (2 \sqrt{21})) (5 (i \cdot \sqrt{7}))$

Step 9

Simplify the expression.

Move $2$ to the left of $i$ .

$8 (2 \cdot i \sqrt{21}) (5 (i \cdot \sqrt{7}))$

Multiply $2$ by $8$ .

$16 (i \sqrt{21}) (5 (i \cdot \sqrt{7}))$

Step 10

Multiply $16 i \sqrt{21} (5 i \sqrt{7})$ .

Multiply $5$ by $16$ .

$80 i \sqrt{21} (i \sqrt{7})$

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

$80 (i^{1} i) \sqrt{21} \sqrt{7}$

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

$80 (i^{1} i^{1}) \sqrt{21} \sqrt{7}$

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

$80 i^{1 + 1} \sqrt{21} \sqrt{7}$

Add $1$ and $1$ .

$80 i^{2} \sqrt{21} \sqrt{7}$

Combine using the product rule for radicals.

$80 i^{2} \sqrt{7 \cdot 21}$

Multiply $7$ by $21$ .

$80 i^{2} \sqrt{147}$

Step 11

Rewrite $i^{2}$ as $- 1$ .

$80 \cdot - 1 \sqrt{147}$

Step 12

Multiply $80$ by $- 1$ .

$- 80 \sqrt{147}$

Step 13

Rewrite $147$ as $7^{2} \cdot 3$ .

Factor $49$ out of $147$ .

$- 80 \sqrt{49 (3)}$

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

$- 80 \sqrt{7^{2} \cdot 3}$

Step 14

Pull terms out from under the radical.

$- 80 (7 \sqrt{3})$

Step 15

Multiply $7$ by $- 80$ .

$- 560 \sqrt{3}$

Step 16

The result can be shown in multiple forms.

Exact Form:

$- 560 \sqrt{3}$

Decimal Form:

$- 969.94845223 \dots$

Simplify (8 square root of -84)(5 square root of -7)

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