1.

Arrange the following in ascending or descending order of magnitude: `root(6)(6)root(3)(7),sqrt(5)`

Answer» `root(4)(6)=6^(1//4),root(3)(7)=7^(1//3),sqrt(5)=r^(1//2)`
LCM of the denominators of the exponents of these three terms, 4,3 and 2 is 12.
Now express the exponent of each term, as a fraction in which the denominator is 12.
`6^((1)/(4))=6^((3)/(12))=(6^(3))^((1)/(12))=root(12)(216)`
`7^((1)/(3))=7^((4)/(12))=(7^(4))^((1)/(12))=root(12)(2401)`
`5^((1)/(2))=5^((6)/(12))=(5^(6))^((1)/(12))=root(12)(15625)`
Now `root(4)(6)=root(12)(216),root(3)(7)=root(12)(2401),sqrt(5)=root(12)(15625)`
Hence , their ascending order is
`root(12)(216),root(12)(2401), root(12)(15623),` i.e., `root(4)(6), root(3)(7), sqrt(5)`
`therefore` The descending order of magnitude of the given radicals is `sqrt(5),root(3)(7), root(4)(6)`.


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