Solve sqrt{1-(32/sqrt{2})^2}=?

$$\sqrt { 1 - ( \frac { 32 } { \sqrt { 2 } } ) ^ { 2 } } = ?$$

Evaluate (complex solution)

$$\sqrt{511}i\approx 22.605309111i$$

Solution Steps

Rationalize the denominator of \frac{32}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\sqrt{1-\left(\frac{32\sqrt{2}}{\left(\sqrt{2}\right)^{2}}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\sqrt{1-\left(\frac{32\sqrt{2}}{2}\right)^{2}}$$

Divide 32\sqrt{2} by 2 to get 16\sqrt{2}.

$$\sqrt{1-\left(16\sqrt{2}\right)^{2}}$$

Expand \left(16\sqrt{2}\right)^{2}.

$$\sqrt{1-16^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate 16 to the power of 2 and get 256.

$$\sqrt{1-256\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\sqrt{1-256\times 2}$$

Multiply 256 and 2 to get 512.

$$\sqrt{1-512}$$

Subtract 512 from 1 to get -511.

$$\sqrt{-511}$$

Factor -511=511\left(-1\right). Rewrite the square root of the product \sqrt{511\left(-1\right)} as the product of square roots \sqrt{511}\sqrt{-1}. By definition, the square root of -1 is i.

$$\sqrt{511}i$$

Evaluate

$$\text{Indeterminate}$$

Solution Steps

Rationalize the denominator of \frac{32}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\sqrt{1-\left(\frac{32\sqrt{2}}{\left(\sqrt{2}\right)^{2}}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\sqrt{1-\left(\frac{32\sqrt{2}}{2}\right)^{2}}$$

Divide 32\sqrt{2} by 2 to get 16\sqrt{2}.

$$\sqrt{1-\left(16\sqrt{2}\right)^{2}}$$

Expand \left(16\sqrt{2}\right)^{2}.

$$\sqrt{1-16^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate 16 to the power of 2 and get 256.

$$\sqrt{1-256\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\sqrt{1-256\times 2}$$

Multiply 256 and 2 to get 512.

$$\sqrt{1-512}$$

Subtract 512 from 1 to get -511.

$$\sqrt{-511}$$

Real Part (complex solution)

$$0$$