Question

simplify: \\(sqrt{-72}\\)
options:
a: \\(-6sqrt{2}\\)
b: \\(3sqrt{8}\\)
c: \\(3sqrt{8}\\)
d: \\(6sqrt{12}\\)
e: \\(6sqrt{2}\\)

Explanation:

Step1: Recall imaginary unit

The square root of a negative number involves the imaginary unit \( i \), where \( i = \sqrt{-1} \). So, \( \sqrt{-72}=\sqrt{72}\times\sqrt{-1}=i\sqrt{72} \).

Step2: Factor 72

Factor 72 into perfect square and remaining factor: \( 72 = 36\times2 \), where 36 is a perfect square (\( 6^2 \)).

Step3: Simplify the square root

Using the property \( \sqrt{ab}=\sqrt{a}\times\sqrt{b} \) (\( a\geq0, b\geq0 \)), we have \( \sqrt{72}=\sqrt{36\times2}=\sqrt{36}\times\sqrt{2}=6\sqrt{2} \).

Step4: Combine with imaginary unit

Substitute back: \( i\sqrt{72}=i\times6\sqrt{2}=6i\sqrt{2} \) (assuming the options have a typo and the "x" is \( i \)). Looking at the options, the one with \( 6\sqrt{2} \) multiplied by the imaginary unit (the "x" symbol) should be the first option (let's assume the first option is \( 6x\sqrt{2} \) where \( x = i \)).

Answer:

Assuming the "x" is the imaginary unit \( i \), the correct option is the first one (let's say the option labeled with the left - most "x" symbol and \( 6\sqrt{2} \) related expression, likely the first option in the visual, e.g., if the first option is A with \( 6x\sqrt{2} \), then A. \( 6x\sqrt{2} \))