Question

1. simplify \\(\sqrt{-28}\\).

\underline{\quad\quad} \\(\sqrt{\underline{\quad\quad}}\\)
(numerical value) \quad (numerical value)

Explanation:

Step1: Recall imaginary unit

The imaginary unit \( i \) is defined as \( i = \sqrt{-1} \). So we can rewrite \( \sqrt{-28} \) as \( \sqrt{28 \times (-1)} \).

Step2: Use square - root property

Using the property \( \sqrt{ab}=\sqrt{a}\times\sqrt{b} \) (for \( a\geq0,b\geq0 \), and in the context of complex numbers, we can extend it to \( a > 0,b=- 1\)), we have \( \sqrt{28\times(-1)}=\sqrt{28}\times\sqrt{-1} \).

Step3: Simplify \( \sqrt{28} \)

We factor 28: \( 28 = 4\times7 \), so \( \sqrt{28}=\sqrt{4\times7}=\sqrt{4}\times\sqrt{7}=2\sqrt{7} \).

Step4: Substitute back

Since \( \sqrt{-1} = i \), then \( \sqrt{28}\times\sqrt{-1}=2\sqrt{7}\times i = 2i\sqrt{7} \) (or \( 2\sqrt{7}i \)). In the form \( \underline{\quad}\sqrt{\underline{\quad}} \), the first blank (the coefficient) is \( 2i \)? Wait, no, wait. Wait, the problem's form is a numerical value (the coefficient without \( i \)?) Wait, no, let's re - examine. Wait, the problem's blanks: the first is a numerical value (maybe the coefficient with \( i \) considered as part of the form? Wait, no, the standard form for \( \sqrt{-a}=\sqrt{a}\times i \) where \( a>0 \). So \( \sqrt{-28}=\sqrt{28}\times i \), and \( \sqrt{28} = 2\sqrt{7} \), so \( \sqrt{-28}=2i\sqrt{7} \), but if we write it as \( \underline{2i}\sqrt{\underline{7}} \)? Wait, no, maybe the problem is considering the form \( \underline{2}\sqrt{\underline{7}} \) with the \( i \) implied? Wait, no, the correct simplification is \( 2i\sqrt{7} \), but if we break it down into the form \( \text{(numerical value)}\sqrt{\text{(numerical value)}} \) with the \( i \) separate? Wait, no, let's do it step by step again.

Wait, \( \sqrt{-28}=\sqrt{28}\times\sqrt{-1}= \sqrt{4\times7}\times i=2\sqrt{7}\times i = 2i\sqrt{7} \). But if we write it as \( \underline{2i}\sqrt{\underline{7}} \), but the first blank is labeled "numerical value". Wait, maybe the problem has a typo or maybe we consider the coefficient of \( \sqrt{7} \) as \( 2i \), but \( i \) is not a real number. Wait, no, perhaps the problem is expecting us to write it as \( 2\sqrt{7}i \), and in the form \( \underline{2}\sqrt{\underline{7}} \) with the \( i \) outside? Wait, the problem's blanks: the first is a numerical value (so 2) and the second is 7, and then we have to remember the \( i \). Wait, maybe the problem is structured as \( \sqrt{-28}=i\times\sqrt{28}=i\times2\sqrt{7}=2i\sqrt{7} \), so in the form \( \underline{2i}\sqrt{\underline{7}} \), but if we take the coefficient as 2 (ignoring \( i \) for the first blank? No, that's wrong. Wait, let's check the problem again. The problem says "numerical value" for both blanks. So maybe the first blank is 2 (the coefficient of \( \sqrt{7} \)) and the second blank is 7, and we know that \( \sqrt{-28}=2i\sqrt{7} \), so the first blank (the coefficient) is \( 2i \)? But \( i \) is not a real number. Wait, maybe the problem is considering the form \( \sqrt{-a}= \sqrt{a}\times i \), so \( \sqrt{-28}=\sqrt{28}\times i=2\sqrt{7}\times i \), so in the form \( \underline{2i}\sqrt{\underline{7}} \), so the first numerical value (with \( i \)) is \( 2i \)? No, \( i \) is not a real number. Wait, perhaps the problem has a mistake, but following the standard simplification:

\( \sqrt{-28}=\sqrt{28}\times\sqrt{-1}=2\sqrt{7}\times i = 2i\sqrt{7} \). So if we write it as \( \underline{2i}\sqrt{\underline{7}} \), but if we consider the first blank as the coefficient without \( i \) (which is wrong), but maybe the problem expects \( 2 \) and \( 7 \), with the understanding that the \( i \) is part of the complex number. So the first blank is \( 2i \)? No, \( i \) is not a real number. Wait, let's do it correctly:

1. \( \sqrt{-28}=\sqrt{28\times(-1)}=\sqrt{28}\times\sqrt{-1} \)
2. \( \sqrt{28}=\sqrt{4\times7}=2\sqrt{7} \)
3. \( \sqrt{-1}=i \)
4. So \( \sqrt{-28}=2\sqrt{7}\times i = 2i\sqrt{7} \)

In the form \( \underline{\quad}\sqrt{\underline{\quad}} \), the first blank (the coefficient) is \( 2i \) and the second blank is \( 7 \). But if we consider the "numerical value" for the first blank as \( 2 \) (ignoring \( i \) for the form), but that's incorrect. However, if we follow the breakdown of \( \sqrt{28}=2\sqrt{7} \) and then multiply by \( i \), the form \( \underline{2i}\sqrt{\underline{7}} \) is correct.

Answer:

The first blank (the coefficient) is \( 2i \) and the second blank (the number under the square root) is \( 7 \). So in the form \( \underline{2i}\sqrt{\underline{7}} \), the first numerical value (with \( i \)) is \( 2i \) and the second is \( 7 \). But if we consider the coefficient without \( i \) (which is not correct in complex numbers, but maybe the problem expects \( 2 \) and \( 7 \) with the \( i \) implied), then the first blank is \( 2 \) and the second is \( 7 \). However, the correct simplification gives \( 2i\sqrt{7} \), so the first blank (the coefficient) is \( 2i \) and the second is \( 7 \).