Question : Which of the following statement(s) is/are true?
I. $\sqrt{12}<\sqrt[3]{16}<\sqrt[4]{24}$
II. $\sqrt[3]{25}>\sqrt[4]{32}>\sqrt[6]{48}$
III. $\sqrt[4]{9}>\sqrt[3]{15}>\sqrt[6]{24}$
Option 1: Only II
Option 2: Only I and III
Option 3: Only I
Option 4: All are true
Correct Answer: Only II
Solution :
By evaluating each statement:
I. $\sqrt{12}<\sqrt[3]{16}<\sqrt[4]{24}$
⇒ $12^\frac{1}{2}<16^\frac{1}{3}<24^\frac{1}{4}$
LCM of 2, 3, and 4 = 12
⇒ $(12^\frac{1}{2})^{12}<(16^\frac{1}{3})^{12}<(24^\frac{1}{4})^{12}$
⇒ $12^6<16^4<24^3$
⇒ $(1728)^2<65536<13824$
So, the first statement is false.
II. $\sqrt[3]{25}>\sqrt[4]{32}>\sqrt[6]{48}$
Similarly,
⇒ $25^4>32^3>48^2$
⇒ $390625>32768>2304$
So, the second statement is true.
III. $\sqrt[4]{9}>\sqrt[3]{15}>\sqrt[6]{24}$
Similarly,
⇒ $9^3>15^4>24^2$
⇒ $729>50625>576$
So, the third statement is false.
Hence, the correct answer is 'Only II'.
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