Write the Fraction in Simplest Form (1+ square root of -81)/( square root of -9* square root of -36)

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

Step 1

Pull out imaginary unit $i$ .

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 $i i$ .

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

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

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

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

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

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

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.

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

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

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

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

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

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

Step 6

Simplify the denominator.

Combine using the product rule for radicals.

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

Multiply $36$ by $9$ .

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

Step 7

Simplify the denominator.

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

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

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.

Move the negative in front of the fraction.

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

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

Factor $9$ out of $9 i$ .

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

Cancel the common factors.

Factor $9$ out of $- 18$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Write the Fraction in Simplest Form (1+ square root of -81)/( square root of -9* square root of -36)

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