Question

express in terms of \\(i\\).

\\(\sqrt{-16}\\)

\\(\sqrt{-16} = \square\\)
(simplify your answer. type your answer in the form \\(a + bi\\).)

Explanation:

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<reasoning>

Define the imaginary unit

We define the imaginary unit \(i\) to represent the square root of \(-1\):
\[i = \sqrt{-1}\]

Apply radical properties

Using the Product Rule for Radicals and Square Root Properties knowledge points
\[\sqrt{-16} = \sqrt{16 \cdot (-1)} = \sqrt{16} \cdot \sqrt{-1}\]

Simplify the expression

Using the Square Root Properties knowledge point
\[\sqrt{16} \cdot \sqrt{-1} = 4 \cdot i = 4i\]

Write in standard complex form

The standard form of a complex number is \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. Since there is no real part, we write:
\[0 + 4i\]
</reasoning>

<answer>
\(\sqrt{-16} =\) <blank>\(0 + 4i\)</blank>
(Simplify your answer. Type your answer in the form a + bi.)
</answer>

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Answer:

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<reasoning>

Define the imaginary unit

We define the imaginary unit \(i\) to represent the square root of \(-1\):
\[i = \sqrt{-1}\]

Apply radical properties

Using the Product Rule for Radicals and Square Root Properties knowledge points
\[\sqrt{-16} = \sqrt{16 \cdot (-1)} = \sqrt{16} \cdot \sqrt{-1}\]

Simplify the expression

Using the Square Root Properties knowledge point
\[\sqrt{16} \cdot \sqrt{-1} = 4 \cdot i = 4i\]

Write in standard complex form

The standard form of a complex number is \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. Since there is no real part, we write:
\[0 + 4i\]
</reasoning>

<answer>
\(\sqrt{-16} =\) <blank>\(0 + 4i\)</blank>
(Simplify your answer. Type your answer in the form a + bi.)
</answer>

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