Question

question 6 of 10
which choice is equivalent to the expression below?
\sqrt{-20}

a. -2\sqrt{5}

b. 20i

c. 2i\sqrt{5}

d. -20i

e. 2\sqrt{5}i

Explanation:

Step1: Recall the imaginary unit

The imaginary unit \( i \) is defined as \( i = \sqrt{-1} \). So we can rewrite \( \sqrt{-20} \) using this definition.
\( \sqrt{-20} = \sqrt{20 \times (-1)} \)

Step2: Use the property of square roots

We know that \( \sqrt{ab} = \sqrt{a} \times \sqrt{b} \) for non - negative real numbers \( a \) and \( b \) (when dealing with complex numbers, we can extend this property). So \( \sqrt{20\times(-1)}=\sqrt{20}\times\sqrt{-1} \)

Step3: Simplify \( \sqrt{20} \)

We can factor \( 20 = 4\times5 \), then \( \sqrt{20}=\sqrt{4\times5}=\sqrt{4}\times\sqrt{5}=2\sqrt{5} \)

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

Since \( \sqrt{-1}=i \), we have \( \sqrt{20}\times\sqrt{-1}=2\sqrt{5}\times i = 2i\sqrt{5} \) (we can also write it as \( 2\sqrt{5}i \), but \( 2i\sqrt{5} \) is in the option C)

Answer:

C. \( 2i\sqrt{5} \)