Sure, let's solve each of these expressions step-by-step to convert them into rational numbers.
### (i) [tex]\(\left(\frac{2}{3}\right)^5\)[/tex]
To raise a fraction to a power, we raise both the numerator and the denominator to the power separately.
[tex]\[
\left(\frac{2}{3}\right)^5 = \frac{2^5}{3^5} = \frac{32}{243}
\][/tex]
This fraction can also be approximated to a decimal:
[tex]\[
\frac{32}{243} \approx 0.13168724279835387
\][/tex]
### (ii) [tex]\(\left(\frac{-8}{5}\right)^3\)[/tex]
Similarly, raise both the numerator and the denominator to the power of 3:
[tex]\[
\left(\frac{-8}{5}\right)^3 = \frac{(-8)^3}{5^3} = \frac{-512}{125}
\][/tex]
[tex]\[
\frac{-512}{125} \approx -4.096
\][/tex]
### (iii) [tex]\(\left(\frac{-13}{11}\right)^2\)[/tex]
Raise both the numerator and the denominator to the power of 2:
[tex]\[
\left(\frac{-13}{11}\right)^2 = \frac{(-13)^2}{11^2} = \frac{169}{121}
\][/tex]
[tex]\[
\frac{169}{121} \approx 1.3966942148760333
\][/tex]
### (v) [tex]\(\left(\frac{-1}{2}\right)^5\)[/tex]
Raise both the numerator and the denominator to the power of 5:
[tex]\[
\left(\frac{-1}{2}\right)^5 = \frac{(-1)^5}{2^5} = \frac{-1}{32}
\][/tex]
[tex]\[
\frac{-1}{32} \approx -0.03125
\][/tex]
### (vi) [tex]\(\left(\frac{-3}{2}\right)^4\)[/tex]
Raise both the numerator and the denominator to the power of 4:
[tex]\[
\left(\frac{-3}{2}\right)^4 = \frac{(-3)^4}{2^4} = \frac{81}{16}
\][/tex]
[tex]\[
\frac{81}{16} \approx 5.0625
\][/tex]
### (vii) [tex]\(\left(\frac{-4}{7}\right)^3\)[/tex]
Raise both the numerator and the denominator to the power of 3:
[tex]\[
\left(\frac{-4}{7}\right)^3 = \frac{(-4)^3}{7^3} = \frac{-64}{343}
\][/tex]
[tex]\[
\frac{-64}{343} \approx -0.1865889212827988
\][/tex]
So, the rational number approximations of each of the expressions are:
1. [tex]\(\left(\frac{2}{3}\right)^5 \approx 0.13168724279835387\)[/tex]
2. [tex]\(\left(\frac{-8}{5}\right)^3 \approx -4.096\)[/tex]
3. [tex]\(\left(\frac{-13}{11}\right)^2 \approx 1.3966942148760333\)[/tex]
5. [tex]\(\left(\frac{-1}{2}\right)^5 \approx -0.03125\)[/tex]
6. [tex]\(\left(\frac{-3}{2}\right)^4 \approx 5.0625\)[/tex]
7. [tex]\(\left(\frac{-4}{7}\right)^3 \approx -0.1865889212827988\)[/tex]
These approximations represent the rational numbers for the given functions.
