Let's solve each of the given expressions step-by-step.
### Expression 1: [tex]\( \frac{1}{\sqrt[7]{2}} \)[/tex]
Interpret [tex]\( \sqrt[7]{2} \)[/tex] as [tex]\( 2^{\frac{1}{7}} \)[/tex]. Therefore, the expression becomes:
[tex]\[ \frac{1}{2^{\frac{1}{7}}} \][/tex]
This evaluates to approximately:
[tex]\[ 0.9057 \][/tex]
### Expression 2: [tex]\( \frac{3}{\sqrt[6]{3^5}} \)[/tex]
First, simplify the denominator:
[tex]\[ \sqrt[6]{3^5} \text{ is equivalent to } (3^5)^{\frac{1}{6}} \Rightarrow 3^{\frac{5}{6}} \][/tex]
So the expression becomes:
[tex]\[ \frac{3}{3^{\frac{5}{6}}} \][/tex]
Which simplifies to:
[tex]\[ 3^{1 - \frac{5}{6}} = 3^{\frac{1}{6}} \][/tex]
This evaluates to approximately:
[tex]\[ 1.2009 \][/tex]
### Expression 3: [tex]\( \frac{-5}{\sqrt[3]{5}} \)[/tex]
Interpret [tex]\( \sqrt[3]{5} \)[/tex] as [tex]\( 5^{\frac{1}{3}} \)[/tex]. Therefore, the expression becomes:
[tex]\[ \frac{-5}{5^{\frac{1}{3}}} \][/tex]
Which simplifies to:
[tex]\[ -5 \cdot 5^{-\frac{1}{3}} = -5^{1 - \frac{1}{3}} = -5^{\frac{2}{3}} \][/tex]
This evaluates to approximately:
[tex]\[ -2.9240 \][/tex]
### Expression 4: [tex]\( \frac{4-\sqrt{5}}{2-\sqrt{13}} \)[/tex]
To find this quotient, calculate the numerator and the denominator separately:
Numerator:
[tex]\[ 4 - \sqrt{5} \][/tex]
Denominator:
[tex]\[ 2 - \sqrt{13} \][/tex]
Then the fraction
[tex]\[ \frac{4 - \sqrt{5}}{2 - \sqrt{13}} \][/tex]
This evaluates to approximately:
[tex]\[ -1.0986 \][/tex]
### Conclusion
By solving each of these expressions, we get the approximate values as follows:
1. [tex]\( \frac{1}{\sqrt[7]{2}} \approx 0.9057 \)[/tex]
2. [tex]\( \frac{3}{\sqrt[6]{3^5}} \approx 1.2009 \)[/tex]
3. [tex]\( \frac{-5}{\sqrt[3]{5}} \approx -2.9240 \)[/tex]
4. [tex]\( \frac{4-\sqrt{5}}{2-\sqrt{13}} \approx -1.0986 \)[/tex]
These are the detailed step-by-step solutions and approximate numerical results for each expression.
