Solve 6/3+sqrt{7}

$$\frac{ 6 }{ 3+ \sqrt{ 7 } }$$

Evaluate

$$9-3\sqrt{7}\approx 1.062746067$$

Solution Steps

Rationalize the denominator of \frac{6}{3+\sqrt{7}} by multiplying numerator and denominator by 3-\sqrt{7}.

$$\frac{6\left(3-\sqrt{7}\right)}{\left(3+\sqrt{7}\right)\left(3-\sqrt{7}\right)}$$

Consider \left(3+\sqrt{7}\right)\left(3-\sqrt{7}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{6\left(3-\sqrt{7}\right)}{3^{2}-\left(\sqrt{7}\right)^{2}}$$

Square 3. Square \sqrt{7}.

$$\frac{6\left(3-\sqrt{7}\right)}{9-7}$$

Subtract 7 from 9 to get 2.

$$\frac{6\left(3-\sqrt{7}\right)}{2}$$

Divide 6\left(3-\sqrt{7}\right) by 2 to get 3\left(3-\sqrt{7}\right).

$$3\left(3-\sqrt{7}\right)$$

Use the distributive property to multiply 3 by 3-\sqrt{7}.

$$9-3\sqrt{7}$$