Question

question 4 of 10
which choice is equivalent to the expression below?
\\(\sqrt{-18}\\)

a. \\(18i\\)
b. \\(3\sqrt{2}i\\)
c. \\(3i\sqrt{2}\\)
d. \\(-18i\\)
e. \\(-3\sqrt{2}\\)

Explanation:

Step1: Recall the imaginary unit

Recall that \( i = \sqrt{-1} \), so we can rewrite \( \sqrt{-18} \) as \( \sqrt{18 \times (-1)} \).

Step2: Use the property of square roots

Using the property \( \sqrt{ab}=\sqrt{a}\times\sqrt{b} \) (for \( a\geq0,b\geq0 \), and here we extend it for complex numbers), we have \( \sqrt{18\times(-1)}=\sqrt{18}\times\sqrt{-1} \).

Step3: Simplify \( \sqrt{18} \)

Simplify \( \sqrt{18} \), we know that \( 18 = 9\times2 \), so \( \sqrt{18}=\sqrt{9\times2}=\sqrt{9}\times\sqrt{2} = 3\sqrt{2} \).

Step4: Substitute \( \sqrt{-1} \) with \( i \)

Since \( \sqrt{-1}=i \), then \( \sqrt{18}\times\sqrt{-1}=3\sqrt{2}\times i=3i\sqrt{2} \).

Answer:

C. \( 3i\sqrt{2} \)