To solve the expression [tex]\(-\sqrt{\frac{-7}{4}} - \sqrt{\frac{-1}{7}}\)[/tex], let's break it down into two parts.
1. First term: [tex]\(-\sqrt{\frac{-7}{4}}\)[/tex]:
- The expression inside the square root is [tex]\(\frac{-7}{4}\)[/tex], which is a negative fraction.
- Taking the square root of a negative number introduces an imaginary unit, [tex]\(i\)[/tex].
- Therefore, [tex]\(\sqrt{\frac{-7}{4}}\)[/tex] can be re-written as [tex]\(\sqrt{\frac{7}{4}} \cdot i\)[/tex].
Now let's calculate [tex]\(\sqrt{\frac{7}{4}}\)[/tex]:
- [tex]\(\sqrt{\frac{7}{4}} = \frac{\sqrt{7}}{\sqrt{4}} = \frac{\sqrt{7}}{2}\)[/tex]
- Thus, [tex]\(\sqrt{\frac{-7}{4}} = \frac{\sqrt{7}}{2} \cdot i\)[/tex]
- Finally, we apply the negative sign: [tex]\(-\sqrt{\frac{-7}{4}} = -\frac{\sqrt{7}}{2} \cdot i\)[/tex]
2. Second term: [tex]\(-\sqrt{\frac{-1}{7}}\)[/tex]:
- The expression inside the square root is [tex]\(\frac{-1}{7}\)[/tex], also a negative fraction.
- As before, the square root of a negative number introduces an imaginary unit, [tex]\(i\)[/tex].
- So [tex]\(\sqrt{\frac{-1}{7}}\)[/tex] becomes [tex]\(\sqrt{\frac{1}{7}} \cdot i\)[/tex].
Now let's calculate [tex]\(\sqrt{\frac{1}{7}}\)[/tex]:
- [tex]\(\sqrt{\frac{1}{7}} = \sqrt{1} / \sqrt{7} = \frac{1}{\sqrt{7}} = \frac{\sqrt{7}}{7}\)[/tex] (by rationalizing the denominator)
- Thus, [tex]\(\sqrt{\frac{-1}{7}} = \frac{\sqrt{7}}{7} \cdot i\)[/tex]
- Finally, we apply the negative sign: [tex]\(-\sqrt{\frac{-1}{7}} = -\frac{\sqrt{7}}{7} \cdot i\)[/tex]
3. Combining the results:
- The first term calculated was [tex]\(-\frac{\sqrt{7}}{2} \cdot i\)[/tex]
- The second term calculated was [tex]\(-\frac{\sqrt{7}}{7} \cdot i\)[/tex]
Bringing both terms together, we have:
[tex]\[ -\frac{\sqrt{7}}{2} \cdot i - \frac{\sqrt{7}}{7} \cdot i \][/tex]
Given the calculated results, the expressions are:
[tex]\[ -\frac{\sqrt{7}}{2} \cdot i \approx (-8.100277186087911e-17 - 1.3228756555322954j) \][/tex]
and
[tex]\[ -\frac{\sqrt{7}}{7} \cdot i \approx (-2.3143649103108314e-17 - 0.3779644730092272j) \][/tex]
The overall result by combining these two terms will be:
[tex]\[ (-8.100277186087911e-17 - 1.3228756555322954j) + (-2.3143649103108314e-17 - 0.3779644730092272j) \][/tex]
[tex]\[ = (-1.0414642096398742e-16 - 1.7008401285415226j) \][/tex]
Hence, the final answer for [tex]\(-\sqrt{\frac{-7}{4}} - \sqrt{\frac{-1}{7}}\)[/tex] is:
[tex]\[ (-1.0414642096398742e-16 - 1.7008401285415226j) \][/tex]
