Simplify 5/(4 square root of -2)

$\frac{5}{4 \sqrt{- 2}}$

Step 1

Pull out imaginary unit $i$ .

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{5}{4 \sqrt{- 1 (2)}}$

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

$\frac{5}{4 (\sqrt{- 1} \cdot \sqrt{2})}$

Rewrite $\sqrt{- 1}$ as $i$ .

$\frac{5}{4 (i \cdot \sqrt{2})}$

Step 2

Multiply $\frac{5}{4 i \cdot \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{5}{4 i \cdot \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 3

Combine and simplify the denominator.

Multiply $\frac{5}{4 i \cdot \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{5 \sqrt{2}}{4 i \cdot \sqrt{2} \sqrt{2}}$

Move $\sqrt{2}$ .

$\frac{5 \sqrt{2}}{4 i (\sqrt{2} \sqrt{2})}$

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{5 \sqrt{2}}{4 i ({\sqrt{2}}^{1} \sqrt{2})}$

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{5 \sqrt{2}}{4 i ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{5 \sqrt{2}}{4 i {\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{5 \sqrt{2}}{4 i {\sqrt{2}}^{2}}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{5 \sqrt{2}}{4 i {(2^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{5 \sqrt{2}}{4 i \cdot 2^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$\frac{5 \sqrt{2}}{4 i \cdot 2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{5 \sqrt{2}}{4 i \cdot 2^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{5 \sqrt{2}}{4 i \cdot 2^{1}}$

Evaluate the exponent.

$\frac{5 \sqrt{2}}{4 i \cdot 2}$

Step 4

Multiply $2$ by $4$ .

$\frac{5 \sqrt{2}}{8 i}$