Question

select all the correct answers.

which expressions are equivalent to \\(\sqrt{-27}\\)?

\\(3i\sqrt{3}\\)
\\(9i\sqrt{3}\\)
\\(\sqrt{27}i\\)
\\(i\sqrt{27}\\)
\\(3\sqrt{9i^2}\\)

Explanation:

Express the radical using the imaginary unit \(i\)

$$ \sqrt{-27} = \sqrt{27} \cdot \sqrt{-1} = \sqrt{27}i = i\sqrt{27} $$

Simplify the square root of 27

$$ \sqrt{27} = \sqrt{9 \cdot 3} = 3\sqrt{3} $$

$$ \sqrt{-27} = 3i\sqrt{3} $$

Evaluate the given options

• \(3i\sqrt{3}\): Equivalent to \(\sqrt{-27}\).
• \(9i\sqrt{3}\): Not equivalent (\(9\sqrt{3}

eq 3\sqrt{3}\)).

• \(\sqrt{27i}\): Not equivalent (the \(i\) is inside the radical).
• \(i\sqrt{27}\): Equivalent to \(\sqrt{-27}\).
• \(3\sqrt{9i^2} = 3\sqrt{-9} = 3 \cdot 3i = 9i\): Not equivalent.

Answer:

• (A) \(3i\sqrt{3}\) (Correct answer)
• (B) \(9i\sqrt{3}\)
• (C) \(\sqrt{27i}\)
• (D) \(i\sqrt{27}\) (Correct answer)
• (E) \(3\sqrt{9i^2}\)