Answer:
The answers are given below.
Step-by-step explanation:
7. Simplified Fractions/Integers:
a. [tex]$ (25)^{\frac{-3}{2}} $[/tex]
= [tex]$ (5^2)^{\frac{-3}{2}} $[/tex]
= [tex]$ 5^{-3} $[/tex]
= [tex]$ \frac{1}{5^3} = \frac{1}{125} $[/tex] is the required answer.
b. [tex]$ \bigg ( \frac{1}{7} \bigg )^{2} $[/tex]
= [tex]$ \bigg( \frac{1}{7^2} \bigg) $[/tex]
= [tex]$ \frac{1}{49} $[/tex] is the required answer.
c. [tex]$ \bigg ( \frac{1}{7} \bigg) ^{-3} $[/tex]
= [tex]$ \bigg ( \frac{1}{7^{-3}} \bigg )$[/tex]
= [tex]$ 7^{3} $[/tex] = 343 is the required answer.
8. Simplified Fractions:
a. [tex]$ \bigg( \frac{27}{8} \bigg ) ^{\frac{2}{3}}$[/tex]
= [tex]$ \bigg ( \frac{3^3}{2^3} \bigg ) ^{\frac{2}{3}} $[/tex]
= [tex]$ \bigg ( \frac{3}{2} \bigg) ^2 $[/tex]
[tex]$ = \frac{9}{4} $[/tex] is the required answer.
b. [tex]$ \bigg ( \frac{4}{25} \bigg)^{\frac{-3}{2}} $[/tex]
= [tex]$ \bigg ( \frac{2^2}{5^2} \bigg)^{\frac{-3}{2}}[/tex]
= [tex]$ \bigg ( \frac{2^{-3}}{5^{-3}} \bigg ) $[/tex]
= [tex]$ \bigg( \frac{5^3}{2^3} \bigg ) $[/tex]
= [tex]$ \frac{125}{8} $[/tex] is the required answer.
9. Index Forms:
a. √3
= [tex]$ 3^{\frac{1}{2}} $[/tex] is the required answer.
b. ∛3
= [tex]$ 3^{\frac{1}{3}} $[/tex] is the required answer.
c. [tex]$\sqrt[6]{2} $[/tex]
= [tex]$ 2^{\frac{1}{6} $[/tex] is the required answer.
d. [tex]$ \sqrt{5^3} $[/tex]
= [tex]$ 5^{\frac{3}{2}} $[/tex] is the required answer.
e. [tex]$ \frac{1}{(\sqrt[3]{5})^2} $[/tex]
= [tex]$ \frac{1}{5^{\frac{2}{3}}} $[/tex]
= [tex]$ 5^{\frac{-2}{3}} $[/tex] is the required answer.
