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