Simplify (3 square root of 7)/(-1- square root of 27)

$\frac{3 \sqrt{7}}{- 1 - \sqrt{27}}$

Step 1

Simplify the denominator.

Rewrite $27$ as $3^{2} \cdot 3$ .

Factor $9$ out of $27$ .

$\frac{3 \sqrt{7}}{- 1 - \sqrt{9 (3)}}$

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

$\frac{3 \sqrt{7}}{- 1 - \sqrt{3^{2} \cdot 3}}$

Pull terms out from under the radical.

$\frac{3 \sqrt{7}}{- 1 - (3 \sqrt{3})}$

Multiply $3$ by $- 1$ .

$\frac{3 \sqrt{7}}{- 1 - 3 \sqrt{3}}$

Step 2

Multiply $\frac{3 \sqrt{7}}{- 1 - 3 \sqrt{3}}$ by $\frac{- 1 + 3 \sqrt{3}}{- 1 + 3 \sqrt{3}}$ .

$\frac{3 \sqrt{7}}{- 1 - 3 \sqrt{3}} \cdot \frac{- 1 + 3 \sqrt{3}}{- 1 + 3 \sqrt{3}}$

Step 3

Multiply $\frac{3 \sqrt{7}}{- 1 - 3 \sqrt{3}}$ by $\frac{- 1 + 3 \sqrt{3}}{- 1 + 3 \sqrt{3}}$ .

$\frac{3 \sqrt{7} (- 1 + 3 \sqrt{3})}{(- 1 - 3 \sqrt{3}) (- 1 + 3 \sqrt{3})}$

Step 4

Expand the denominator using the FOIL method.

$\frac{3 \sqrt{7} (- 1 + 3 \sqrt{3})}{1 - 3 \sqrt{3} + 3 \sqrt{3} - 9 {\sqrt{3}}^{2}}$

Step 5

Simplify.

$\frac{3 \sqrt{7} (- 1 + 3 \sqrt{3})}{- 26}$

Step 6

Group $- 1 + 3 \sqrt{3}$ and $\sqrt{7}$ together.

$\frac{3 ((- 1 + 3 \sqrt{3}) \sqrt{7})}{- 26}$

Step 7

Apply the distributive property.

$\frac{3 (- 1 \sqrt{7} + 3 \sqrt{3} \sqrt{7})}{- 26}$

Step 8

Rewrite $- 1 \sqrt{7}$ as $- \sqrt{7}$ .

$\frac{3 (- \sqrt{7} + 3 \sqrt{3} \sqrt{7})}{- 26}$

Step 9

Multiply $3 \sqrt{3} \sqrt{7}$ .

Combine using the product rule for radicals.

$\frac{3 (- \sqrt{7} + 3 \sqrt{7 \cdot 3})}{- 26}$

Multiply $7$ by $3$ .

$\frac{3 (- \sqrt{7} + 3 \sqrt{21})}{- 26}$

Step 10

Move the negative in front of the fraction.

$- \frac{3 (- \sqrt{7} + 3 \sqrt{21})}{26}$

Step 11

Factor $- 1$ out of $- \sqrt{7}$ .

$- \frac{3 (- (\sqrt{7}) + 3 \sqrt{21})}{26}$

Step 12

Factor $- 1$ out of $3 \sqrt{21}$ .

$- \frac{3 (- (\sqrt{7}) - (- 3 \sqrt{21}))}{26}$

Step 13

Factor $- 1$ out of $- (\sqrt{7}) - (- 3 \sqrt{21})$ .

$- \frac{3 (- (\sqrt{7} - 3 \sqrt{21}))}{26}$

Step 14

Simplify the expression.

Rewrite $- (\sqrt{7} - 3 \sqrt{21})$ as $- 1 (\sqrt{7} - 3 \sqrt{21})$ .

$- \frac{3 (- 1 (\sqrt{7} - 3 \sqrt{21}))}{26}$

Move the negative in front of the fraction.

$- - \frac{3 (\sqrt{7} - 3 \sqrt{21})}{26}$

Multiply $- 1$ by $- 1$ .

$1 \frac{3 (\sqrt{7} - 3 \sqrt{21})}{26}$

Multiply $\frac{3 (\sqrt{7} - 3 \sqrt{21})}{26}$ by $1$ .

$\frac{3 (\sqrt{7} - 3 \sqrt{21})}{26}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$\frac{3 (\sqrt{7} - 3 \sqrt{21})}{26}$

Decimal Form:

$- 1.28099720 \dots$

Simplify (3 square root of 7)/(-1- square root of 27)

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