Answer:
A. [tex](\sqrt[3]{125})^9\ and\ (125)^{\frac{9}{3}}[/tex]
D. [tex]8^{\frac{9}{2}}\ and\ (\sqrt{8})^9[/tex]
Step-by-step explanation:
Equivalent expressions are those expressions that simplify to same form.
Now, let us check each of the given options.
Option A:
[tex](\sqrt[3]{125})^9\ and\ (125)^{\frac{9}{3}}[/tex]
We know that,
[tex]\sqrt[n]{x} =x^{\frac{1}{n}}[/tex]
Therefore, [tex]\sqrt[3]{125} =(125)^{\frac{1}{3}}[/tex]
Thus the first expression becomes;
[tex]((125)^{\frac{1}{3}})^9[/tex]
Now, using law of indices [tex](a^m)^n=a^{m\times n}[/tex], we get
[tex]((125)^{\frac{1}{3}})^9=((125))^{\frac{1}{3}\times 9}=((125))^{\frac{9}{3}[/tex]
Therefore, [tex](\sqrt[3]{125})^9\ and\ (125)^{\frac{9}{3}}[/tex] are equivalent.
Option B:
[tex]12^{\frac{2}{7}}\ and\ (\sqrt{12})^7[/tex]
Consider the second expression [tex](\sqrt{12})^7[/tex]
We know that,
[tex]\sqrt x=x^{\frac{1}{2}}[/tex]
[tex](\sqrt{12})^7=((12)^{\frac{1}{2}})^7=(12)^{\frac{1}{2}\times 7}=(12)^{\frac{7}{2}}[/tex]
Therefore, [tex]12^{\frac{2}{7}} \ne (12)^{\frac{7}{2}}[/tex]. Hence, the expressions [tex]12^{\frac{2}{7}}\ and\ (\sqrt{12})^7[/tex] are not equivalent.
Option C:
[tex]4^{\frac{1}{5}}\ and\ (\sqrt 4)^5[/tex]
We know that,
[tex]x^{\frac{1}{n}}=\sqrt[n]{x}[/tex]
Therefore, [tex]4^{\frac{1}{5}}=\sqrt[5]{4}[/tex]
Now, [tex]\sqrt[5]{4}\ne (\sqrt 4)^5[/tex]
Therefore, the expressions [tex]4^{\frac{1}{5}}\ and\ (\sqrt 4)^5[/tex] are not equivalent.
Option D:
[tex]8^{\frac{9}{2} }\ and\ (\sqrt 8)^9[/tex]
Using law of indices [tex]a^{m\times n}=(a^m)^n[/tex], we get
[tex]8^{\frac{9}{2} }=(8^{\frac{1}{2}})^9[/tex]
Now, we know that, [tex]x^{\frac{1}{2}}=\sqrt x[/tex]
So, [tex](8^{\frac{1}{2}})^9=(\sqrt8)^9[/tex]
Therefore, [tex]8^{\frac{9}{2} }\ and\ (\sqrt 8)^9[/tex] are equivalent.
