Simplify 2/(6+3 square root of 5)

$\frac{2}{6 + 3 \sqrt{5}}$

Step 1

Multiply $\frac{2}{6 + 3 \sqrt{5}}$ by $\frac{6 - 3 \sqrt{5}}{6 - 3 \sqrt{5}}$ .

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

Step 2

Multiply $\frac{2}{6 + 3 \sqrt{5}}$ by $\frac{6 - 3 \sqrt{5}}{6 - 3 \sqrt{5}}$ .

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

Step 3

Expand the denominator using the FOIL method.

$\frac{2 (6 - 3 \sqrt{5})}{36 - 18 \sqrt{5} + 18 \sqrt{5} - 9 {\sqrt{5}}^{2}}$

Step 4

Simplify.

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

Step 5

Cancel the common factor of $6 - 3 \sqrt{5}$ and $- 9$ .

Factor $3$ out of $2 (6 - 3 \sqrt{5})$ .

$\frac{3 (2 (2 - \sqrt{5}))}{- 9}$

Cancel the common factors.

Factor $3$ out of $- 9$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 6

Move the negative in front of the fraction.

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.15737865 \dots$