Algebra Examples

$\frac{4 + \sqrt{27}}{2 - 3 \sqrt{27}}$

Step 1

Simplify the numerator.

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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 $27$ .

$\frac{4 + \sqrt{9 (3)}}{2 - 3 \sqrt{27}}$

Step 1.1.2

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

$\frac{4 + \sqrt{3^{2} \cdot 3}}{2 - 3 \sqrt{27}}$

Step 1.2

Pull terms out from under the radical.

$\frac{4 + 3 \sqrt{3}}{2 - 3 \sqrt{27}}$

Step 2

Simplify the denominator.

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Step 2.1

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

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Step 2.1.1

Factor $9$ out of $27$ .

$\frac{4 + 3 \sqrt{3}}{2 - 3 \sqrt{9 (3)}}$

Step 2.1.2

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

$\frac{4 + 3 \sqrt{3}}{2 - 3 \sqrt{3^{2} \cdot 3}}$

Step 2.2

Pull terms out from under the radical.

$\frac{4 + 3 \sqrt{3}}{2 - 3 (3 \sqrt{3})}$

Step 2.3

Multiply $3$ by $- 3$ .

$\frac{4 + 3 \sqrt{3}}{2 - 9 \sqrt{3}}$

Step 3

Multiply $\frac{4 + 3 \sqrt{3}}{2 - 9 \sqrt{3}}$ by $\frac{2 + 9 \sqrt{3}}{2 + 9 \sqrt{3}}$ .

$\frac{4 + 3 \sqrt{3}}{2 - 9 \sqrt{3}} \cdot \frac{2 + 9 \sqrt{3}}{2 + 9 \sqrt{3}}$

Step 4

Combine fractions.

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Step 4.1

Multiply $\frac{4 + 3 \sqrt{3}}{2 - 9 \sqrt{3}}$ by $\frac{2 + 9 \sqrt{3}}{2 + 9 \sqrt{3}}$ .

$\frac{(4 + 3 \sqrt{3}) (2 + 9 \sqrt{3})}{(2 - 9 \sqrt{3}) (2 + 9 \sqrt{3})}$

Step 4.2

Expand the denominator using the FOIL method.

$\frac{(4 + 3 \sqrt{3}) (2 + 9 \sqrt{3})}{4 + 18 \sqrt{3} - 18 \sqrt{3} - 81 {\sqrt{3}}^{2}}$

Step 4.3

Simplify.

$\frac{(4 + 3 \sqrt{3}) (2 + 9 \sqrt{3})}{- 239}$

Step 5

Expand $(4 + 3 \sqrt{3}) (2 + 9 \sqrt{3})$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$\frac{4 (2 + 9 \sqrt{3}) + 3 \sqrt{3} (2 + 9 \sqrt{3})}{- 239}$

Step 5.2

Apply the distributive property.

$\frac{4 \cdot 2 + 4 (9 \sqrt{3}) + 3 \sqrt{3} (2 + 9 \sqrt{3})}{- 239}$

Step 5.3

Apply the distributive property.

$\frac{4 \cdot 2 + 4 (9 \sqrt{3}) + 3 \sqrt{3} \cdot 2 + 3 \sqrt{3} (9 \sqrt{3})}{- 239}$

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $4$ by $2$ .

$\frac{8 + 4 (9 \sqrt{3}) + 3 \sqrt{3} \cdot 2 + 3 \sqrt{3} (9 \sqrt{3})}{- 239}$

Step 6.1.2

Multiply $9$ by $4$ .

$\frac{8 + 36 \sqrt{3} + 3 \sqrt{3} \cdot 2 + 3 \sqrt{3} (9 \sqrt{3})}{- 239}$

Step 6.1.3

Multiply $2$ by $3$ .

$\frac{8 + 36 \sqrt{3} + 6 \sqrt{3} + 3 \sqrt{3} (9 \sqrt{3})}{- 239}$

Step 6.1.4

Multiply $3 \sqrt{3} (9 \sqrt{3})$ .

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Step 6.1.4.1

Multiply $9$ by $3$ .

$\frac{8 + 36 \sqrt{3} + 6 \sqrt{3} + 27 \sqrt{3} \sqrt{3}}{- 239}$

Step 6.1.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{8 + 36 \sqrt{3} + 6 \sqrt{3} + 27 ({\sqrt{3}}^{1} \sqrt{3})}{- 239}$

Step 6.1.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{8 + 36 \sqrt{3} + 6 \sqrt{3} + 27 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}{- 239}$

Step 6.1.4.4

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

$\frac{8 + 36 \sqrt{3} + 6 \sqrt{3} + 27 {\sqrt{3}}^{1 + 1}}{- 239}$

Step 6.1.4.5

Add $1$ and $1$ .

$\frac{8 + 36 \sqrt{3} + 6 \sqrt{3} + 27 {\sqrt{3}}^{2}}{- 239}$

Step 6.1.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 6.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{8 + 36 \sqrt{3} + 6 \sqrt{3} + 27 {(3^{\frac{1}{2}})}^{2}}{- 239}$

Step 6.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{8 + 36 \sqrt{3} + 6 \sqrt{3} + 27 \cdot 3^{\frac{1}{2} \cdot 2}}{- 239}$

Step 6.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{8 + 36 \sqrt{3} + 6 \sqrt{3} + 27 \cdot 3^{\frac{2}{2}}}{- 239}$

Step 6.1.5.4

Cancel the common factor of $2$ .

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Step 6.1.5.4.1

Cancel the common factor.

$\frac{8 + 36 \sqrt{3} + 6 \sqrt{3} + 27 \cdot 3^{\frac{2}{2}}}{- 239}$

Step 6.1.5.4.2

Rewrite the expression.

$\frac{8 + 36 \sqrt{3} + 6 \sqrt{3} + 27 \cdot 3^{1}}{- 239}$

Step 6.1.5.5

Evaluate the exponent.

$\frac{8 + 36 \sqrt{3} + 6 \sqrt{3} + 27 \cdot 3}{- 239}$

Step 6.1.6

Multiply $27$ by $3$ .

$\frac{8 + 36 \sqrt{3} + 6 \sqrt{3} + 81}{- 239}$

Step 6.2

Add $8$ and $81$ .

$\frac{89 + 36 \sqrt{3} + 6 \sqrt{3}}{- 239}$

Step 6.3

Add $36 \sqrt{3}$ and $6 \sqrt{3}$ .

$\frac{89 + 42 \sqrt{3}}{- 239}$

Step 7

Move the negative in front of the fraction.

$- \frac{89 + 42 \sqrt{3}}{239}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{89 + 42 \sqrt{3}}{239}$

Decimal Form:

$- 0.67676206 \dots$