Simplify the following
`(20)/(sqrt(-5))`

Answer

To simplify the expression \( \frac{20}{\sqrt{-5}} \), we can follow these steps:

### Step 1: Rewrite the square root of a negative number
We know that \( \sqrt{-5} \) can be expressed as \( \sqrt{5} \cdot \sqrt{-1} \). Since \( \sqrt{-1} = i \) (where \( i \) is the imaginary unit), we can rewrite the expression:

\[ \sqrt{-5} = \sqrt{5} \cdot i \]

### Step 2: Substitute back into the expression
Now, we substitute this back into the original expression:

\[ \frac{20}{\sqrt{-5}} = \frac{20}{\sqrt{5} \cdot i} \]

### Step 3: Multiply numerator and denominator by \( i \)
To eliminate the imaginary unit from the denominator, we multiply both the numerator and the denominator by \( i \):

\[ \frac{20 \cdot i}{\sqrt{5} \cdot i^2} \]

Since \( i^2 = -1 \), we can simplify the denominator:

\[ \frac{20 \cdot i}{\sqrt{5} \cdot (-1)} = \frac{-20i}{\sqrt{5}} \]

### Step 4: Simplify the expression
Now, we can express the fraction in a more standard form:

\[ \frac{-20i}{\sqrt{5}} = -\frac{20}{\sqrt{5}} i \]

### Step 5: Rationalize the denominator
To rationalize the denominator, we multiply the numerator and denominator by \( \sqrt{5} \):

\[ -\frac{20}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = -\frac{20\sqrt{5}}{5} \]

### Step 6: Simplify the fraction
Now, we simplify the fraction:

\[ -\frac{20\sqrt{5}}{5} = -4\sqrt{5} \]

### Final Answer
Thus, the simplified form of \( \frac{20}{\sqrt{-5}} \) is:

\[ -4\sqrt{5} i \]

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