Solve P(1/sqrt{2}-5/2sqrt{2})^-2

$$P ( \frac { 1 } { \sqrt { 2 } } - \frac { 5 } { 2 \sqrt { 2 } } ) ^ { - 2 }$$

Evaluate

$$\frac{8P}{9}$$

Solution Steps

Rationalize the denominator of \frac{1}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$P\left(\frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}-\frac{5}{2\sqrt{2}}\right)^{-2}$$

The square of \sqrt{2} is 2.

$$P\left(\frac{\sqrt{2}}{2}-\frac{5}{2\sqrt{2}}\right)^{-2}$$

Rationalize the denominator of \frac{5}{2\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$P\left(\frac{\sqrt{2}}{2}-\frac{5\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}\right)^{-2}$$

The square of \sqrt{2} is 2.

$$P\left(\frac{\sqrt{2}}{2}-\frac{5\sqrt{2}}{2\times 2}\right)^{-2}$$

Multiply 2 and 2 to get 4.

$$P\left(\frac{\sqrt{2}}{2}-\frac{5\sqrt{2}}{4}\right)^{-2}$$

Combine \frac{\sqrt{2}}{2} and -\frac{5\sqrt{2}}{4} to get -\frac{3}{4}\sqrt{2}.

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

Expand \left(-\frac{3}{4}\sqrt{2}\right)^{-2}.

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

Calculate -\frac{3}{4} to the power of -2 and get \frac{16}{9}.

$$P\times \frac{16}{9}\left(\sqrt{2}\right)^{-2}$$

Expand

$$\frac{8P}{9}$$

Solution Steps

Rationalize the denominator of \frac{1}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$P\left(\frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}-\frac{5}{2\sqrt{2}}\right)^{-2}$$

The square of \sqrt{2} is 2.

$$P\left(\frac{\sqrt{2}}{2}-\frac{5}{2\sqrt{2}}\right)^{-2}$$

Rationalize the denominator of \frac{5}{2\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$P\left(\frac{\sqrt{2}}{2}-\frac{5\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}\right)^{-2}$$

The square of \sqrt{2} is 2.

$$P\left(\frac{\sqrt{2}}{2}-\frac{5\sqrt{2}}{2\times 2}\right)^{-2}$$

Multiply 2 and 2 to get 4.

$$P\left(\frac{\sqrt{2}}{2}-\frac{5\sqrt{2}}{4}\right)^{-2}$$

Combine \frac{\sqrt{2}}{2} and -\frac{5\sqrt{2}}{4} to get -\frac{3}{4}\sqrt{2}.

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

Expand \left(-\frac{3}{4}\sqrt{2}\right)^{-2}.

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

Calculate -\frac{3}{4} to the power of -2 and get \frac{16}{9}.

$$P\times \frac{16}{9}\left(\sqrt{2}\right)^{-2}$$