What is `sqrt(-18)timessqrt(-50)` written in simplest form? (Note: `i=sqrt(-1)`)

Options

A. `-30`
B. `30i`
C. `30`
D. `30i`

Answer

To simplify the expression \( \sqrt{-18} \times \sqrt{-50} \), we can follow these steps:

### Step 1: Rewrite the square roots
We start by rewriting the square roots of negative numbers using the imaginary unit \( i \), where \( i = \sqrt{-1} \).

\[ \sqrt{-18} = \sqrt{-1 \times 18} = \sqrt{-1} \times \sqrt{18} = i \sqrt{18} \]
\[ \sqrt{-50} = \sqrt{-1 \times 50} = \sqrt{-1} \times \sqrt{50} = i \sqrt{50} \]

### Step 2: Multiply the expressions
Now, we can multiply the two expressions together:

\[ \sqrt{-18} \times \sqrt{-50} = (i \sqrt{18}) \times (i \sqrt{50}) \]

### Step 3: Combine the imaginary units
When we multiply \( i \) by \( i \), we get \( i^2 \):

\[ (i \sqrt{18}) \times (i \sqrt{50}) = i^2 \times \sqrt{18} \times \sqrt{50} \]

### Step 4: Substitute \( i^2 \) with \(-1\)
Since \( i^2 = -1 \), we can substitute this into our expression:

\[ i^2 \times \sqrt{18} \times \sqrt{50} = -1 \times \sqrt{18} \times \sqrt{50} \]

### Step 5: Simplify the square roots
Now we can combine the square roots:

\[ \sqrt{18} \times \sqrt{50} = \sqrt{18 \times 50} = \sqrt{900} \]

### Step 6: Calculate the square root
Now we can find the square root of 900:

\[ \sqrt{900} = 30 \]

### Step 7: Final expression
Putting it all together, we have:

\[ -1 \times \sqrt{900} = -1 \times 30 = -30 \]

Thus, the simplest form of \( \sqrt{-18} \times \sqrt{-50} \) is:

\[ \boxed{-30} \]

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