Solve 21/sqrt{2-4}

$$\frac{ 21 }{ \sqrt{ 2 } -4 }$$

Evaluate

$$-\frac{3\sqrt{2}}{2}-6\approx -8.121320344$$

Solution Steps

Rationalize the denominator of \frac{21}{\sqrt{2}-4} by multiplying numerator and denominator by \sqrt{2}+4.

$$\frac{21\left(\sqrt{2}+4\right)}{\left(\sqrt{2}-4\right)\left(\sqrt{2}+4\right)}$$

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

$$\frac{21\left(\sqrt{2}+4\right)}{\left(\sqrt{2}\right)^{2}-4^{2}}$$

Square \sqrt{2}. Square 4.

$$\frac{21\left(\sqrt{2}+4\right)}{2-16}$$

Subtract 16 from 2 to get -14.

$$\frac{21\left(\sqrt{2}+4\right)}{-14}$$

Divide 21\left(\sqrt{2}+4\right) by -14 to get -\frac{3}{2}\left(\sqrt{2}+4\right).

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

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

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

Express -\frac{3}{2}\times 4 as a single fraction.

$$-\frac{3}{2}\sqrt{2}+\frac{-3\times 4}{2}$$

Multiply -3 and 4 to get -12.

$$-\frac{3}{2}\sqrt{2}+\frac{-12}{2}$$

Divide -12 by 2 to get -6.

$$-\frac{3}{2}\sqrt{2}-6$$