Algebra Examples

$(- 3 \sqrt{- 81}) (- 5 + \sqrt{- 9}) + 9 i$

Step 1

Simplify each term.

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Step 1.1

Rewrite $- 81$ as $- 1 (81)$ .

$- 3 \sqrt{- 1 (81)} (- 5 + \sqrt{- 9}) + 9 i$

Step 1.2

Rewrite $\sqrt{- 1 (81)}$ as $\sqrt{- 1} \cdot \sqrt{81}$ .

$- 3 (\sqrt{- 1} \cdot \sqrt{81}) (- 5 + \sqrt{- 9}) + 9 i$

Step 1.3

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

$- 3 (i \cdot \sqrt{81}) (- 5 + \sqrt{- 9}) + 9 i$

Step 1.4

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

$- 3 (i \cdot \sqrt{81}) (- 5 + \sqrt{- 1 (9)}) + 9 i$

Step 1.5

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

$- 3 (i \cdot \sqrt{81}) (- 5 + \sqrt{- 1} \cdot \sqrt{9}) + 9 i$

Step 1.6

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

$- 3 (i \cdot \sqrt{81}) (- 5 + i \cdot \sqrt{9}) + 9 i$

Step 1.7

Rewrite $81$ as $9^{2}$ .

$- 3 (i \cdot \sqrt{9^{2}}) (- 5 + i \cdot \sqrt{9}) + 9 i$

Step 1.8

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

$- 3 (i \cdot 9) (- 5 + i \cdot \sqrt{9}) + 9 i$

Step 1.9

Move $9$ to the left of $i$ .

$- 3 (9 \cdot i) (- 5 + i \cdot \sqrt{9}) + 9 i$

Step 1.10

Multiply $9$ by $- 3$ .

$- 27 i (- 5 + i \cdot \sqrt{9}) + 9 i$

Step 1.11

Simplify each term.

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Step 1.11.1

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

$- 27 i (- 5 + i \cdot \sqrt{3^{2}}) + 9 i$

Step 1.11.2

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

$- 27 i (- 5 + i \cdot 3) + 9 i$

Step 1.11.3

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

$- 27 i (- 5 + 3 i) + 9 i$

Step 1.12

Apply the distributive property.

$- 27 i \cdot - 5 - 27 i (3 i) + 9 i$

Step 1.13

Multiply $- 5$ by $- 27$ .

$135 i - 27 i (3 i) + 9 i$

Step 1.14

Multiply $- 27 i (3 i)$ .

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Step 1.14.1

Multiply $3$ by $- 27$ .

$135 i - 81 i i + 9 i$

Step 1.14.2

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

$135 i - 81 (i^{1} i) + 9 i$

Step 1.14.3

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

$135 i - 81 (i^{1} i^{1}) + 9 i$

Step 1.14.4

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

$135 i - 81 i^{1 + 1} + 9 i$

Step 1.14.5

Add $1$ and $1$ .

$135 i - 81 i^{2} + 9 i$

Step 1.15

Simplify each term.

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Step 1.15.1

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

$135 i - 81 \cdot - 1 + 9 i$

Step 1.15.2

Multiply $- 81$ by $- 1$ .

$135 i + 81 + 9 i$

Step 2

Add $135 i$ and $9 i$ .

$81 + 144 i$