Simplify (2- square root of 5)/(2+3 square root of 5)

$\frac{2 - \sqrt{5}}{2 + 3 \sqrt{5}}$

Step 1

Multiply $\frac{2 - \sqrt{5}}{2 + 3 \sqrt{5}}$ by $\frac{2 - 3 \sqrt{5}}{2 - 3 \sqrt{5}}$ .

$\frac{2 - \sqrt{5}}{2 + 3 \sqrt{5}} \cdot \frac{2 - 3 \sqrt{5}}{2 - 3 \sqrt{5}}$

Step 2

Combine fractions.

Multiply $\frac{2 - \sqrt{5}}{2 + 3 \sqrt{5}}$ by $\frac{2 - 3 \sqrt{5}}{2 - 3 \sqrt{5}}$ .

$\frac{(2 - \sqrt{5}) (2 - 3 \sqrt{5})}{(2 + 3 \sqrt{5}) (2 - 3 \sqrt{5})}$

Expand the denominator using the FOIL method.

$\frac{(2 - \sqrt{5}) (2 - 3 \sqrt{5})}{4 - 6 \sqrt{5} + 6 \sqrt{5} - 9 {\sqrt{5}}^{2}}$

Simplify.

$\frac{(2 - \sqrt{5}) (2 - 3 \sqrt{5})}{- 41}$

Step 3

Expand $(2 - \sqrt{5}) (2 - 3 \sqrt{5})$ using the FOIL Method.

Apply the distributive property.

$\frac{2 (2 - 3 \sqrt{5}) - \sqrt{5} (2 - 3 \sqrt{5})}{- 41}$

Apply the distributive property.

$\frac{2 \cdot 2 + 2 (- 3 \sqrt{5}) - \sqrt{5} (2 - 3 \sqrt{5})}{- 41}$

Apply the distributive property.

$\frac{2 \cdot 2 + 2 (- 3 \sqrt{5}) - \sqrt{5} \cdot 2 - \sqrt{5} (- 3 \sqrt{5})}{- 41}$

Step 4

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

$\frac{4 + 2 (- 3 \sqrt{5}) - \sqrt{5} \cdot 2 - \sqrt{5} (- 3 \sqrt{5})}{- 41}$

Multiply $- 3$ by $2$ .

$\frac{4 - 6 \sqrt{5} - \sqrt{5} \cdot 2 - \sqrt{5} (- 3 \sqrt{5})}{- 41}$

Multiply $2$ by $- 1$ .

$\frac{4 - 6 \sqrt{5} - 2 \sqrt{5} - \sqrt{5} (- 3 \sqrt{5})}{- 41}$

Multiply $- \sqrt{5} (- 3 \sqrt{5})$ .

Multiply $- 3$ by $- 1$ .

$\frac{4 - 6 \sqrt{5} - 2 \sqrt{5} + 3 \sqrt{5} \sqrt{5}}{- 41}$

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{4 - 6 \sqrt{5} - 2 \sqrt{5} + 3 ({\sqrt{5}}^{1} \sqrt{5})}{- 41}$

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{4 - 6 \sqrt{5} - 2 \sqrt{5} + 3 ({\sqrt{5}}^{1} {\sqrt{5}}^{1})}{- 41}$

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

$\frac{4 - 6 \sqrt{5} - 2 \sqrt{5} + 3 {\sqrt{5}}^{1 + 1}}{- 41}$

Add $1$ and $1$ .

$\frac{4 - 6 \sqrt{5} - 2 \sqrt{5} + 3 {\sqrt{5}}^{2}}{- 41}$

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{4 - 6 \sqrt{5} - 2 \sqrt{5} + 3 {(5^{\frac{1}{2}})}^{2}}{- 41}$

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

$\frac{4 - 6 \sqrt{5} - 2 \sqrt{5} + 3 \cdot 5^{\frac{1}{2} \cdot 2}}{- 41}$

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

$\frac{4 - 6 \sqrt{5} - 2 \sqrt{5} + 3 \cdot 5^{\frac{2}{2}}}{- 41}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{4 - 6 \sqrt{5} - 2 \sqrt{5} + 3 \cdot 5^{\frac{2}{2}}}{- 41}$

Rewrite the expression.

$\frac{4 - 6 \sqrt{5} - 2 \sqrt{5} + 3 \cdot 5^{1}}{- 41}$

Evaluate the exponent.

$\frac{4 - 6 \sqrt{5} - 2 \sqrt{5} + 3 \cdot 5}{- 41}$

Multiply $3$ by $5$ .

$\frac{4 - 6 \sqrt{5} - 2 \sqrt{5} + 15}{- 41}$

Add $4$ and $15$ .

$\frac{19 - 6 \sqrt{5} - 2 \sqrt{5}}{- 41}$

Subtract $2 \sqrt{5}$ from $- 6 \sqrt{5}$ .

$\frac{19 - 8 \sqrt{5}}{- 41}$

Step 5

Move the negative in front of the fraction.

$- \frac{19 - 8 \sqrt{5}}{41}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{19 - 8 \sqrt{5}}{41}$

Decimal Form:

$- 0.02710868 \dots$

Simplify (2- square root of 5)/(2+3 square root of 5)

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