Question

question 2 of 10
which choice is equivalent to the expression below?
\\(\sqrt{-27}\\)

a. \\(3i\sqrt{3}\\)
b. \\(-3\sqrt{3}\\)
c. \\(3\sqrt{3}\\)
d. \\(-3\sqrt{3}i\\)
e. \\(-\sqrt{27}\\)

Explanation:

Step1: Recall the imaginary unit

The imaginary unit \( i \) is defined as \( i = \sqrt{-1} \), so we can rewrite \( \sqrt{-27} \) as \( \sqrt{27 \times (-1)} \).

Step2: Use the property of square roots

Using the property \( \sqrt{ab} = \sqrt{a} \times \sqrt{b} \) (for \( a\geq0, b\geq0 \), and extended to complex numbers here), we have \( \sqrt{27 \times (-1)}=\sqrt{27}\times\sqrt{-1} \).

Step3: Simplify \( \sqrt{27} \)

We know that \( 27 = 9\times3 \), so \( \sqrt{27}=\sqrt{9\times3}=\sqrt{9}\times\sqrt{3}=3\sqrt{3} \).

Step4: Substitute back

Since \( \sqrt{-1} = i \), we substitute \( \sqrt{27}=3\sqrt{3} \) and \( \sqrt{-1}=i \) into \( \sqrt{27}\times\sqrt{-1} \), we get \( 3\sqrt{3}\times i = 3i\sqrt{3} \).

Answer:

A. \( 3i\sqrt{3} \)