Solve -3/3+2sqrt{7}

$$\frac { - 3 } { 3 + 2 \sqrt { 7 } }$$

Evaluate

$$\frac{9-6\sqrt{7}}{19}\approx -0.361816203$$

Solution Steps

Rationalize the denominator of \frac{-3}{3+2\sqrt{7}} by multiplying numerator and denominator by 3-2\sqrt{7}.

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

Consider \left(3+2\sqrt{7}\right)\left(3-2\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}.

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

Calculate 3 to the power of 2 and get 9.

$$\frac{-3\left(3-2\sqrt{7}\right)}{9-\left(2\sqrt{7}\right)^{2}}$$

Expand \left(2\sqrt{7}\right)^{2}.

$$\frac{-3\left(3-2\sqrt{7}\right)}{9-2^{2}\left(\sqrt{7}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{-3\left(3-2\sqrt{7}\right)}{9-4\left(\sqrt{7}\right)^{2}}$$

The square of \sqrt{7} is 7.

$$\frac{-3\left(3-2\sqrt{7}\right)}{9-4\times 7}$$

Multiply 4 and 7 to get 28.

$$\frac{-3\left(3-2\sqrt{7}\right)}{9-28}$$

Subtract 28 from 9 to get -19.

$$\frac{-3\left(3-2\sqrt{7}\right)}{-19}$$

Use the distributive property to multiply -3 by 3-2\sqrt{7}.

$$\frac{-9+6\sqrt{7}}{-19}$$