Basic Math Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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Step 1.1.1

Factor $9$ out of $27$ .

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

Step 1.1.2

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

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

Step 1.2

Pull terms out from under the radical.

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

Step 1.3

Multiply $3$ by $- 1$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Expand the denominator using the FOIL method.

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

Step 5

Simplify.

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

Step 6

Apply the distributive property.

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

Step 7

Move $- 1$ to the left of $\sqrt{3 \sqrt{7}}$ .

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

Step 8

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

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Step 8.1

Combine using the product rule for radicals.

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

Step 8.2

Multiply $3$ by $3$ .

$\frac{- 1 \cdot \sqrt{3 \sqrt{7}} + 3 \sqrt{9 \sqrt{7}}}{- 26}$

Step 9

Simplify each term.

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Step 9.1

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

$\frac{- \sqrt{3 \sqrt{7}} + 3 \sqrt{9 \sqrt{7}}}{- 26}$

Step 9.2

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

$\frac{- \sqrt{3 \sqrt{7}} + 3 \sqrt{3^{2} \sqrt{7}}}{- 26}$

Step 9.3

Pull terms out from under the radical.

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

Step 9.4

Rewrite $\sqrt{\sqrt{7}}$ as $\sqrt[4]{7}$ .

$\frac{- \sqrt{3 \sqrt{7}} + 3 (3 \sqrt[4]{7})}{- 26}$

Step 9.5

Multiply $3$ by $3$ .

$\frac{- \sqrt{3 \sqrt{7}} + 9 \sqrt[4]{7}}{- 26}$

Step 10

Simplify with factoring out.

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Step 10.1

Move the negative in front of the fraction.

$- \frac{- \sqrt{3 \sqrt{7}} + 9 \sqrt[4]{7}}{26}$

Step 10.2

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

$- \frac{- (\sqrt{3 \sqrt{7}}) + 9 \sqrt[4]{7}}{26}$

Step 10.3

Factor $- 1$ out of $9 \sqrt[4]{7}$ .

$- \frac{- (\sqrt{3 \sqrt{7}}) - (- 9 \sqrt[4]{7})}{26}$

Step 10.4

Factor $- 1$ out of $- (\sqrt{3 \sqrt{7}}) - (- 9 \sqrt[4]{7})$ .

$- \frac{- (\sqrt{3 \sqrt{7}} - 9 \sqrt[4]{7})}{26}$

Step 10.5

Simplify the expression.

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Step 10.5.1

Rewrite $- (\sqrt{3 \sqrt{7}} - 9 \sqrt[4]{7})$ as $- 1 (\sqrt{3 \sqrt{7}} - 9 \sqrt[4]{7})$ .

$- \frac{- 1 (\sqrt{3 \sqrt{7}} - 9 \sqrt[4]{7})}{26}$

Step 10.5.2

Move the negative in front of the fraction.

$- - \frac{\sqrt{3 \sqrt{7}} - 9 \sqrt[4]{7}}{26}$

Step 10.5.3

Multiply $- 1$ by $- 1$ .

$1 \frac{\sqrt{3 \sqrt{7}} - 9 \sqrt[4]{7}}{26}$

Step 10.5.4

Multiply $\frac{\sqrt{3 \sqrt{7}} - 9 \sqrt[4]{7}}{26}$ by $1$ .

$\frac{\sqrt{3 \sqrt{7}} - 9 \sqrt[4]{7}}{26}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{3 \sqrt{7}} - 9 \sqrt[4]{7}}{26}$

Decimal Form:

$- 0.45468753 \dots$