Answer:
C and D
Step-by-step explanation:
using the rules of exponents
A
[tex]125^{3/7}[/tex] = [tex]\sqrt[7]{125^3}[/tex] ≠ [tex]\sqrt[3]{125^7}[/tex]
B
([tex]\sqrt{12}[/tex])^7 = [tex]12^{7/2}[/tex] ≠ [tex]12^{1/7}[/tex]
C
([tex]\sqrt{4}[/tex])^5 = [tex]4^{5/2}[/tex] ← correct
D
([tex]\sqrt{8}[/tex])^9 = [tex]8^{9/2}[/tex] ← correct
Answer: The correct options are
(C) [tex]4^\frac{5}{2}~~\textup{and}~~(\sqrt{4})^5.[/tex]
(D) [tex]8^\frac{9}{2}~~\textup{and}~~(\sqrt{8})^9.[/tex]
Step-by-step explanation: We are to select the correct pairs that shows equivalent expressions.
We will be using the following property of exponents and radicals :
[tex](\sqrt[b]{x})^a=x^\frac{a}{b}.[/tex]
Option (A) :
The given expressions are
[tex](\sqrt[3]{125})^7~~\textup{and}~~125^\frac{3}{7}.[/tex]
We have
[tex](\sqrt[3]{125})^7=125^\frac{7}{3}\neq 125^\frac{3}{7}.[/tex]
So, the expressions are not equivalent and option (A) is incorrect.
Option (B) :
The given expressions are
[tex]12^\frac{1}{7}~~\textup{and}~~(\sqrt{12})^7.[/tex]
We have
[tex]12^\frac{1}{7}=\sqrt[7]{12},\\\\(\sqrt{12})^7=12^\frac{7}{2}.[/tex]
So,
[tex]12^\frac{1}{7}\neq (\sqrt{12})^7.[/tex]
Therefore, the expressions are not equivalent and option (B) is incorrect.
Option (C) :
The given expressions are
[tex]4^\frac{5}{2}~~\textup{and}~~(\sqrt{4})^5.[/tex]
We have
[tex](\sqrt{4})^5=4^\frac{5}{2}[/tex]
Therefore, the expressions are equivalent and option (C) is correct.
Option (D) :
The given expressions are
[tex]8^\frac{9}{2}~~\textup{and}~~(\sqrt{8})^9.[/tex]
We have
[tex](\sqrt{8})^9=8^\frac{9}{2}[/tex]
Therefore, the expressions are equivalent and option (D) is correct.
Thus, (C) and (D) are the correct options.
