Solve frac{2(4-sqrt{7})}{sqrt{7}+4}

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

Evaluate

$$\frac{46-16\sqrt{7}}{9}\approx 0.407553225$$

Solution Steps

Rationalize the denominator of \frac{2\left(4-\sqrt{7}\right)}{\sqrt{7}+4} by multiplying numerator and denominator by \sqrt{7}-4.

$$\frac{2\left(4-\sqrt{7}\right)\left(\sqrt{7}-4\right)}{\left(\sqrt{7}+4\right)\left(\sqrt{7}-4\right)}$$

Consider \left(\sqrt{7}+4\right)\left(\sqrt{7}-4\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square \sqrt{7}. Square 4.

$$\frac{2\left(4-\sqrt{7}\right)\left(\sqrt{7}-4\right)}{7-16}$$

Subtract 16 from 7 to get -9.

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

Use the distributive property to multiply 2 by 4-\sqrt{7}.

$$\frac{\left(8-2\sqrt{7}\right)\left(\sqrt{7}-4\right)}{-9}$$

Apply the distributive property by multiplying each term of 8-2\sqrt{7} by each term of \sqrt{7}-4.

$$\frac{8\sqrt{7}-32-2\left(\sqrt{7}\right)^{2}+8\sqrt{7}}{-9}$$

The square of \sqrt{7} is 7.

$$\frac{8\sqrt{7}-32-2\times 7+8\sqrt{7}}{-9}$$

Multiply -2 and 7 to get -14.

$$\frac{8\sqrt{7}-32-14+8\sqrt{7}}{-9}$$

Subtract 14 from -32 to get -46.

$$\frac{8\sqrt{7}-46+8\sqrt{7}}{-9}$$

Combine 8\sqrt{7} and 8\sqrt{7} to get 16\sqrt{7}.

$$\frac{16\sqrt{7}-46}{-9}$$