Simplify 4/(3+ square root of 7)

$\frac{4}{3 + \sqrt{7}}$

Step 1

Multiply $\frac{4}{3 + \sqrt{7}}$ by $\frac{3 - \sqrt{7}}{3 - \sqrt{7}}$ .

$\frac{4}{3 + \sqrt{7}} \cdot \frac{3 - \sqrt{7}}{3 - \sqrt{7}}$

Step 2

Multiply $\frac{4}{3 + \sqrt{7}}$ by $\frac{3 - \sqrt{7}}{3 - \sqrt{7}}$ .

$\frac{4 (3 - \sqrt{7})}{(3 + \sqrt{7}) (3 - \sqrt{7})}$

Step 3

Expand the denominator using the FOIL method.

$\frac{4 (3 - \sqrt{7})}{9 - 3 \sqrt{7} + \sqrt{7} \cdot 3 - {\sqrt{7}}^{2}}$

Step 4

Simplify.

$\frac{4 (3 - \sqrt{7})}{2}$

Step 5

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4 (3 - \sqrt{7})$ .

$\frac{2 (2 (3 - \sqrt{7}))}{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$\frac{2 (2 (3 - \sqrt{7}))}{2 (1)}$

Cancel the common factor.

$\frac{2 (2 (3 - \sqrt{7}))}{2 \cdot 1}$

Rewrite the expression.

$\frac{2 (3 - \sqrt{7})}{1}$

Divide $2 (3 - \sqrt{7})$ by $1$ .

$2 (3 - \sqrt{7})$

Step 6

Apply the distributive property.

$2 \cdot 3 + 2 (- \sqrt{7})$

Step 7

Multiply.

Multiply $2$ by $3$ .

$6 + 2 (- \sqrt{7})$

Multiply $- 1$ by $2$ .

$6 - 2 \sqrt{7}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$6 - 2 \sqrt{7}$

Decimal Form:

$0.70849737 \dots$