Algebra Examples

$\frac{1 + \log (3)}{2 + \sqrt{5}}$

Step 1

Multiply $\frac{1 + \log (3)}{2 + \sqrt{5}}$ by $\frac{2 - \sqrt{5}}{2 - \sqrt{5}}$ .

$\frac{1 + \log (3)}{2 + \sqrt{5}} \cdot \frac{2 - \sqrt{5}}{2 - \sqrt{5}}$

Step 2

Combine fractions.

Tap for more steps...

Step 2.1

Multiply $\frac{1 + \log (3)}{2 + \sqrt{5}}$ by $\frac{2 - \sqrt{5}}{2 - \sqrt{5}}$ .

$\frac{(1 + \log (3)) (2 - \sqrt{5})}{(2 + \sqrt{5}) (2 - \sqrt{5})}$

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

$\frac{(1 + \log (3)) (2 - \sqrt{5})}{- 1}$

Step 2.4

Simplify the expression.

Tap for more steps...

Step 2.4.1

Move the negative one from the denominator of $\frac{(1 + \log (3)) (2 - \sqrt{5})}{- 1}$ .

$- 1 \cdot ((1 + \log (3)) (2 - \sqrt{5}))$

Step 2.4.2

Rewrite $- 1 \cdot ((1 + \log (3)) (2 - \sqrt{5}))$ as $- ((1 + \log (3)) (2 - \sqrt{5}))$ .

$- ((1 + \log (3)) (2 - \sqrt{5}))$

Step 3

Expand $(1 + \log (3)) (2 - \sqrt{5})$ using the FOIL Method.

Tap for more steps...

Step 3.1

Apply the distributive property.

$- (1 (2 - \sqrt{5}) + \log (3) (2 - \sqrt{5}))$

Step 3.2

Apply the distributive property.

$- (1 \cdot 2 + 1 (- \sqrt{5}) + \log (3) (2 - \sqrt{5}))$

Step 3.3

Apply the distributive property.

$- (1 \cdot 2 + 1 (- \sqrt{5}) + \log (3) \cdot 2 + \log (3) (- \sqrt{5}))$

Step 4

Simplify terms.

Tap for more steps...

Step 4.1

Simplify each term.

Tap for more steps...

Step 4.1.1

Multiply $2$ by $1$ .

$- (2 + 1 (- \sqrt{5}) + \log (3) \cdot 2 + \log (3) (- \sqrt{5}))$

Step 4.1.2

Multiply $- \sqrt{5}$ by $1$ .

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

Step 4.1.3

Multiply $\log (3) \cdot 2$ .

Tap for more steps...

Step 4.1.3.1

Reorder $\log (3)$ and $2$ .

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

Step 4.1.3.2

Simplify $2 \cdot \log (3)$ by moving $2$ inside the logarithm.

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

Step 4.1.4

Raise $3$ to the power of $2$ .

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

Step 4.2

Apply the distributive property.

$- 1 \cdot 2 - - \sqrt{5} - \log (9) - (\log (3) (- \sqrt{5}))$

Step 5

Simplify.

Tap for more steps...

Step 5.1

Multiply $- 1$ by $2$ .

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

Step 5.2

Multiply $- - \sqrt{5}$ .

Tap for more steps...

Step 5.2.1

Multiply $- 1$ by $- 1$ .

$- 2 + 1 \sqrt{5} - \log (9) - (\log (3) (- \sqrt{5}))$

Step 5.2.2

Multiply $\sqrt{5}$ by $1$ .

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

Step 5.3

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

Tap for more steps...

Step 5.3.1

Multiply $- 1$ by $- 1$ .

$- 2 + \sqrt{5} - \log (9) + 1 (\log (3) (\sqrt{5}))$

Step 5.3.2

Multiply $\log (3)$ by $1$ .

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.34870102 \dots$