Solve 8/3sqrt{2-4}

$$\frac { 8 } { 3 \sqrt { 2 } - 4 }$$

Evaluate

$$12\sqrt{2}+16\approx 32.970562748$$

Solution Steps

Rationalize the denominator of \frac{8}{3\sqrt{2}-4} by multiplying numerator and denominator by 3\sqrt{2}+4.

$$\frac{8\left(3\sqrt{2}+4\right)}{\left(3\sqrt{2}-4\right)\left(3\sqrt{2}+4\right)}$$

Consider \left(3\sqrt{2}-4\right)\left(3\sqrt{2}+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}.

$$\frac{8\left(3\sqrt{2}+4\right)}{\left(3\sqrt{2}\right)^{2}-4^{2}}$$

Expand \left(3\sqrt{2}\right)^{2}.

$$\frac{8\left(3\sqrt{2}+4\right)}{3^{2}\left(\sqrt{2}\right)^{2}-4^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{8\left(3\sqrt{2}+4\right)}{9\left(\sqrt{2}\right)^{2}-4^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{8\left(3\sqrt{2}+4\right)}{9\times 2-4^{2}}$$

Multiply 9 and 2 to get 18.

$$\frac{8\left(3\sqrt{2}+4\right)}{18-4^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{8\left(3\sqrt{2}+4\right)}{18-16}$$

Subtract 16 from 18 to get 2.

$$\frac{8\left(3\sqrt{2}+4\right)}{2}$$

Divide 8\left(3\sqrt{2}+4\right) by 2 to get 4\left(3\sqrt{2}+4\right).

$$4\left(3\sqrt{2}+4\right)$$

Use the distributive property to multiply 4 by 3\sqrt{2}+4.

$$12\sqrt{2}+16$$