Simplify 1/(2 square root of 5+2)-1/(2 square root of 5-2)

$\frac{1}{2 \sqrt{5} + 2} - \frac{1}{2 \sqrt{5} - 2}$

Simplify each term.

Multiply $\frac{1}{2 \sqrt{5} + 2}$ by $\frac{2 \sqrt{5} - 2}{2 \sqrt{5} - 2}$ .

$\frac{1}{2 \sqrt{5} + 2} \cdot \frac{2 \sqrt{5} - 2}{2 \sqrt{5} - 2} - \frac{1}{2 \sqrt{5} - 2}$

Multiply $\frac{1}{2 \sqrt{5} + 2}$ by $\frac{2 \sqrt{5} - 2}{2 \sqrt{5} - 2}$ .

$\frac{2 \sqrt{5} - 2}{(2 \sqrt{5} + 2) (2 \sqrt{5} - 2)} - \frac{1}{2 \sqrt{5} - 2}$

Expand the denominator using the FOIL method.

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

Simplify.

$\frac{2 \sqrt{5} - 2}{16} - \frac{1}{2 \sqrt{5} - 2}$

Cancel the common factor of $2 \sqrt{5} - 2$ and $16$ .

Factor $2$ out of $2 \sqrt{5}$ .

$\frac{2 (\sqrt{5}) - 2}{16} - \frac{1}{2 \sqrt{5} - 2}$

Factor $2$ out of $- 2$ .

$\frac{2 (\sqrt{5}) + 2 (- 1)}{16} - \frac{1}{2 \sqrt{5} - 2}$

Factor $2$ out of $2 (\sqrt{5}) + 2 (- 1)$ .

$\frac{2 (\sqrt{5} - 1)}{16} - \frac{1}{2 \sqrt{5} - 2}$

Cancel the common factors.

Factor $2$ out of $16$ .

$\frac{2 (\sqrt{5} - 1)}{2 (8)} - \frac{1}{2 \sqrt{5} - 2}$

Cancel the common factor.

$\frac{2 (\sqrt{5} - 1)}{2 \cdot 8} - \frac{1}{2 \sqrt{5} - 2}$

Rewrite the expression.

$\frac{\sqrt{5} - 1}{8} - \frac{1}{2 \sqrt{5} - 2}$

Multiply $\frac{1}{2 \sqrt{5} - 2}$ by $\frac{2 \sqrt{5} + 2}{2 \sqrt{5} + 2}$ .

$\frac{\sqrt{5} - 1}{8} - (\frac{1}{2 \sqrt{5} - 2} \cdot \frac{2 \sqrt{5} + 2}{2 \sqrt{5} + 2})$

Multiply $\frac{1}{2 \sqrt{5} - 2}$ by $\frac{2 \sqrt{5} + 2}{2 \sqrt{5} + 2}$ .

$\frac{\sqrt{5} - 1}{8} - \frac{2 \sqrt{5} + 2}{(2 \sqrt{5} - 2) (2 \sqrt{5} + 2)}$

Expand the denominator using the FOIL method.

$\frac{\sqrt{5} - 1}{8} - \frac{2 \sqrt{5} + 2}{4 {\sqrt{5}}^{2} + 4 \sqrt{5} - 4 \sqrt{5} - 4}$

Simplify.

$\frac{\sqrt{5} - 1}{8} - \frac{2 \sqrt{5} + 2}{16}$

Cancel the common factor of $2 \sqrt{5} + 2$ and $16$ .

Factor $2$ out of $2 \sqrt{5}$ .

$\frac{\sqrt{5} - 1}{8} - \frac{2 (\sqrt{5}) + 2}{16}$

Factor $2$ out of $2$ .

$\frac{\sqrt{5} - 1}{8} - \frac{2 (\sqrt{5}) + 2 (1)}{16}$

Factor $2$ out of $2 (\sqrt{5}) + 2 (1)$ .

$\frac{\sqrt{5} - 1}{8} - \frac{2 (\sqrt{5} + 1)}{16}$

Cancel the common factors.

Factor $2$ out of $16$ .

$\frac{\sqrt{5} - 1}{8} - \frac{2 (\sqrt{5} + 1)}{2 (8)}$

Cancel the common factor.

$\frac{\sqrt{5} - 1}{8} - \frac{2 (\sqrt{5} + 1)}{2 \cdot 8}$

Rewrite the expression.

$\frac{\sqrt{5} - 1}{8} - \frac{\sqrt{5} + 1}{8}$

Combine the numerators over the common denominator.

$\frac{\sqrt{5} - 1 - (\sqrt{5} + 1)}{8}$

Simplify each term.

Apply the distributive property.

$\frac{\sqrt{5} - 1 - \sqrt{5} - 1 \cdot 1}{8}$

Multiply $- 1$ by $1$ .

$\frac{\sqrt{5} - 1 - \sqrt{5} - 1}{8}$

Simplify terms.

Subtract $\sqrt{5}$ from $\sqrt{5}$ .

$\frac{0 - 1 - 1}{8}$

Simplify by subtracting numbers.

Subtract $1$ from $0$ .

$\frac{- 1 - 1}{8}$

Subtract $1$ from $- 1$ .

$\frac{- 2}{8}$

Cancel the common factor of $- 2$ and $8$ .

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{8}$

Cancel the common factors.

Factor $2$ out of $8$ .

$\frac{2 \cdot - 1}{2 \cdot 4}$

Cancel the common factor.

$\frac{2 \cdot - 1}{2 \cdot 4}$

Rewrite the expression.

$\frac{- 1}{4}$

Move the negative in front of the fraction.

$- \frac{1}{4}$

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{4}$

Decimal Form:

$- 0.25$

Simplify 1/(2 square root of 5+2)-1/(2 square root of 5-2)

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