Questions: Simplify the expressions below as much as possible. Leave no negative numbers under radicals and no radicals in the denominator sqrt(-14) * sqrt(-2) = sqrt(-4) / sqrt(2) =

Solution Steps

1. For the expression \(\sqrt{-14} \cdot \sqrt{-2}\), recognize that the square root of a negative number involves imaginary numbers. Use the property \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\) and simplify using \(i\) where \(i = \sqrt{-1}\).

2. For the expression \(\frac{\sqrt{-4}}{\sqrt{2}}\), simplify the square roots separately, convert the negative square root to involve \(i\), and then simplify the fraction.

To simplify \(\sqrt{-14} \cdot \sqrt{-2}\), we first recognize that the square root of a negative number involves the imaginary unit \(i\), where \(i = \sqrt{-1}\). Thus, we can express the square roots as:

\[ \sqrt{-14} = \sqrt{14} \cdot i \quad \text{and} \quad \sqrt{-2} = \sqrt{2} \cdot i \]

Using the property \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\), we have:

\[ \sqrt{-14} \cdot \sqrt{-2} = (\sqrt{14} \cdot i) \cdot (\sqrt{2} \cdot i) = \sqrt{28} \cdot i^2 \]

Since \(i^2 = -1\), this simplifies to:

\[ \sqrt{28} \cdot (-1) = -\sqrt{28} \]

Simplifying \(\sqrt{28}\), we get:

\[ \sqrt{28} = \sqrt{4 \cdot 7} = 2\sqrt{7} \]

Thus, the expression simplifies to:

\[ -2\sqrt{7} \approx -5.2915 \]

For the expression \(\frac{\sqrt{-4}}{\sqrt{2}}\), we first simplify the square roots:

\[ \sqrt{-4} = \sqrt{4} \cdot i = 2i \]

\[ \sqrt{2} = \sqrt{2} \]

Thus, the expression becomes:

\[ \frac{2i}{\sqrt{2}} = \frac{2}{\sqrt{2}} \cdot i \]

Simplifying \(\frac{2}{\sqrt{2}}\), we have:

\[ \frac{2}{\sqrt{2}} = \sqrt{2} \]

Therefore, the expression simplifies to:

\[ \sqrt{2} \cdot i \approx 1.4142i \]

Final Answer