Find the Complex Conjugate 3/(4+4 square root of 5)

$\frac{3}{4 + 4 \sqrt{5}}$

Multiply $\frac{3}{4 + 4 \sqrt{5}}$ by $\frac{4 - 4 \sqrt{5}}{4 - 4 \sqrt{5}}$ .

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

Multiply $\frac{3}{4 + 4 \sqrt{5}}$ by $\frac{4 - 4 \sqrt{5}}{4 - 4 \sqrt{5}}$ .

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

Expand the denominator using the FOIL method.

$\frac{3 (4 - 4 \sqrt{5})}{16 - 16 \sqrt{5} + 16 \sqrt{5} - 16 {\sqrt{5}}^{2}}$

Simplify.

$\frac{3 (4 - 4 \sqrt{5})}{- 64}$

Cancel the common factor of $4 - 4 \sqrt{5}$ and $- 64$ .

Factor $4$ out of $3 (4 - 4 \sqrt{5})$ .

$\frac{4 (3 (1 - \sqrt{5}))}{- 64}$

Cancel the common factors.

Factor $4$ out of $- 64$ .

$\frac{4 (3 (1 - \sqrt{5}))}{4 (- 16)}$

Cancel the common factor.

$\frac{4 (3 (1 - \sqrt{5}))}{4 \cdot - 16}$

Rewrite the expression.

$\frac{3 (1 - \sqrt{5})}{- 16}$

Move the negative in front of the fraction.

$- \frac{3 (1 - \sqrt{5})}{16}$

Since there are no imaginary terms, the complex conjugate is the same as the simplified expression.

$- \frac{3 (1 - \sqrt{5})}{16}$

The result can be shown in multiple forms.

Exact Form:

$- \frac{3 (1 - \sqrt{5})}{16}$

Decimal Form:

$0.23176274 \dots$