Algebra Examples

$\frac{1 + \sqrt{- 81}}{\sqrt{- 9} \cdot \sqrt{- 36}}$

Step 1

Pull out imaginary unit $i$ .

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Step 1.1

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

$\frac{1 + \sqrt{- 1 (81)}}{\sqrt{- 9} \cdot \sqrt{- 36}}$

Step 1.2

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

$\frac{1 + \sqrt{- 1} \cdot \sqrt{81}}{\sqrt{- 9} \cdot \sqrt{- 36}}$

Step 1.3

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

$\frac{1 + i \cdot \sqrt{81}}{\sqrt{- 9} \cdot \sqrt{- 36}}$

Step 1.4

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

$\frac{1 + i \cdot \sqrt{81}}{\sqrt{- 1 (9)} \cdot \sqrt{- 36}}$

Step 1.5

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

$\frac{1 + i \cdot \sqrt{81}}{\sqrt{- 1} \cdot \sqrt{9} \cdot \sqrt{- 36}}$

Step 1.6

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

$\frac{1 + i \cdot \sqrt{81}}{i \cdot \sqrt{9} \cdot \sqrt{- 36}}$

Step 1.7

Rewrite $- 36$ as $- 1 (36)$ .

$\frac{1 + i \cdot \sqrt{81}}{i \cdot \sqrt{9} \cdot \sqrt{- 1 (36)}}$

Step 1.8

Rewrite $\sqrt{- 1 (36)}$ as $\sqrt{- 1} \cdot \sqrt{36}$ .

$\frac{1 + i \cdot \sqrt{81}}{i \cdot \sqrt{9} \cdot (\sqrt{- 1} \cdot \sqrt{36})}$

Step 1.9

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

$\frac{1 + i \cdot \sqrt{81}}{i \cdot \sqrt{9} \cdot (i \cdot \sqrt{36})}$

Step 2

Group $i$ and $i$ together.

$\frac{1 + i \cdot \sqrt{81}}{i i \cdot \sqrt{9} \cdot \sqrt{36}}$

Step 3

Multiply $i i$ .

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Step 3.1

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

$\frac{1 + i \cdot \sqrt{81}}{i^{1} i \cdot \sqrt{9} \cdot \sqrt{36}}$

Step 3.2

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

$\frac{1 + i \cdot \sqrt{81}}{i^{1} i^{1} \cdot \sqrt{9} \cdot \sqrt{36}}$

Step 3.3

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

$\frac{1 + i \cdot \sqrt{81}}{i^{1 + 1} \cdot \sqrt{9} \cdot \sqrt{36}}$

Step 3.4

Add $1$ and $1$ .

$\frac{1 + i \cdot \sqrt{81}}{i^{2} \cdot \sqrt{9} \cdot \sqrt{36}}$

Step 4

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

$\frac{1 + i \cdot \sqrt{81}}{- 1 \cdot \sqrt{9} \cdot \sqrt{36}}$

Step 5

Simplify the numerator.

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Step 5.1

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

$\frac{1 + i \cdot \sqrt{9^{2}}}{- 1 \cdot \sqrt{9} \cdot \sqrt{36}}$

Step 5.2

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

$\frac{1 + i \cdot 9}{- 1 \cdot \sqrt{9} \cdot \sqrt{36}}$

Step 5.3

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

$\frac{1 + 9 i}{- 1 \cdot \sqrt{9} \cdot \sqrt{36}}$

Step 6

Simplify the denominator.

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Step 6.1

Combine using the product rule for radicals.

$\frac{1 + 9 i}{- 1 \cdot \sqrt{36 \cdot 9}}$

Step 6.2

Multiply $36$ by $9$ .

$\frac{1 + 9 i}{- 1 \cdot \sqrt{324}}$

Step 7

Simplify the denominator.

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Step 7.1

Rewrite $324$ as $18^{2}$ .

$\frac{1 + 9 i}{- 1 \sqrt{18^{2}}}$

Step 7.2

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

$\frac{1 + 9 i}{- 1 \cdot 18}$

Step 8

Multiply $- 1$ by $18$ .

$\frac{1 + 9 i}{- 18}$

Step 9

Split the fraction $\frac{1 + 9 i}{- 18}$ into two fractions.

$\frac{1}{- 18} + \frac{9 i}{- 18}$

Step 10

Simplify each term.

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Step 10.1

Move the negative in front of the fraction.

$- \frac{1}{18} + \frac{9 i}{- 18}$

Step 10.2

Cancel the common factor of $9$ and $- 18$ .

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Step 10.2.1

Factor $9$ out of $9 i$ .

$- \frac{1}{18} + \frac{9 (i)}{- 18}$

Step 10.2.2

Cancel the common factors.

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Step 10.2.2.1

Factor $9$ out of $- 18$ .

$- \frac{1}{18} + \frac{9 i}{9 \cdot - 2}$

Step 10.2.2.2

Cancel the common factor.

$- \frac{1}{18} + \frac{9 i}{9 \cdot - 2}$

Step 10.2.2.3

Rewrite the expression.

$- \frac{1}{18} + \frac{i}{- 2}$

Step 10.3

Move the negative in front of the fraction.

$- \frac{1}{18} - \frac{i}{2}$