Question : The greatest among the following numbers $(3)^{\frac{1}{3}}, (2)^{\frac{1}{2}}, 1, (6)^{\frac{1}{6}}$ is:
Option 1: $(2)^{\frac{1}{2}}$
Option 2: 1
Option 3: $(6)^{\frac{1}{6}}$
Option 4: $(3)^{\frac{1}{3}}$
Correct Answer: $(3)^{\frac{1}{3}}$
Solution :
To find the greatest number, we have to find the LCM of the denominators of the power.
Given numbers are = $(3)^{\frac{1}{3}},(2)^{\frac{1}{2}}, 1, (6)^{\frac{1}{6}}$
The LCM of 3, 2, 1 and 6 = 6
Raised the power 6 to each of the numbers.
$⇒ [(3)^{\frac{1}{3}}]^6 = (3)^{2} = 9$
$⇒ [(2)^{\frac{1}{2}}]^6 = (2)^{3} = 8$
$⇒ (1)^{6} = 1$
$⇒ [(6)^{\frac{1}{6}}]^6 = (6)^{1} = 6$
So, $(3)^{\frac{1}{3}}$ is the greatest number.
Hence, the correct answer is $(3)^{\frac{1}{3}}$.
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