Simplify (7 square root of -9)(- square root of -80)

$(7 \sqrt{- 9}) (- \sqrt{- 80})$

Step 1

Rewrite $- 9$ as $- 1 (9)$ .

$7 \sqrt{- 1 (9)} (- \sqrt{- 80})$

Step 2

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$7 (\sqrt{- 1} \cdot \sqrt{9}) (- \sqrt{- 80})$

Step 3

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

$7 (i \cdot \sqrt{9}) (- \sqrt{- 80})$

Step 4

Rewrite $- 80$ as $- 1 (80)$ .

$7 (i \cdot \sqrt{9}) (- \sqrt{- 1 (80)})$

Step 5

Rewrite $\sqrt{- 1 (80)}$ as $\sqrt{- 1} \cdot \sqrt{80}$ .

$7 (i \cdot \sqrt{9}) (- (\sqrt{- 1} \cdot \sqrt{80}))$

Step 6

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

$7 (i \cdot \sqrt{9}) (- (i \cdot \sqrt{80}))$

Step 7

Rewrite $9$ as $3^{2}$ .

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

Step 8

Pull terms out from under the radical, assuming positive real numbers.

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

Step 9

Move $3$ to the left of $i$ .

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

Step 10

Multiply $3$ by $7$ .

$21 i (- (i \cdot \sqrt{80}))$

Step 11

Rewrite $80$ as $4^{2} \cdot 5$ .

Factor $16$ out of $80$ .

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

Rewrite $16$ as $4^{2}$ .

$21 i (- (i \cdot \sqrt{4^{2} \cdot 5}))$

Step 12

Pull terms out from under the radical.

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

Step 13

Simplify the expression.

Move $4$ to the left of $i$ .

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

Multiply $4$ by $- 1$ .

$21 i (- 4 (i \sqrt{5}))$

Step 14

Multiply $21 i (- 4 i \sqrt{5})$ .

Multiply $- 4$ by $21$ .

$- 84 i (i \sqrt{5})$

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

$- 84 (i^{1} i) \sqrt{5}$

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

$- 84 (i^{1} i^{1}) \sqrt{5}$

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

$- 84 i^{1 + 1} \sqrt{5}$

Add $1$ and $1$ .

$- 84 i^{2} \sqrt{5}$

Step 15

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

$- 84 \cdot - 1 \sqrt{5}$

Step 16

Multiply $- 84$ by $- 1$ .

$84 \sqrt{5}$

Step 17

The result can be shown in multiple forms.

Exact Form:

$84 \sqrt{5}$

Decimal Form:

$187.82971010 \dots$

Simplify (7 square root of -9)(- square root of -80)

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