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Maths-

General

Easy

Question

Which is greater ?
$root index 6 of 3 comma space fourth root of 5 comma space square root of 2$

hint Hint:

Use LCM property

The correct answer is: Use LCM property

Complete step by step solution:
Here we can write, $square root of 2 equals 2 to the power of 1 half end exponent comma root index 6 of 3 equals 3 to the power of 1 over 6 end exponent$ and $fourth root of 5 equals 5 to the power of 1 fourth end exponent$
On taking the LCM of 2,6,4 we have LCM as 12.
So, $not stretchy rightwards double arrow 2 to the power of 1 half end exponent equals 2 to the power of fraction numerator 1 cross times 6 over denominator 2 cross times 6 end fraction end exponent equals 2 to the power of 6 over 12 end exponent$
$not stretchy rightwards double arrow open parentheses 2 to the power of 6 close parentheses to the power of 1 over 12 end exponent equals 64 to the power of 1 over 12 end exponent$ (Since ) $open parentheses a to the power of b close parentheses to the power of c equals a to the power of b c end exponent$
Likewise, $not stretchy rightwards double arrow 3 to the power of 1 over 6 end exponent equals 3 to the power of fraction numerator 1 cross times 2 over denominator 6 cross times 2 end fraction end exponent equals 3 to the power of 2 over 12 end exponent$
$not stretchy rightwards double arrow open parentheses 3 squared close parentheses to the power of 1 over 12 end exponent equals 9 to the power of 1 over 12 end exponent$ (Since $open open parentheses a to the power of b close parentheses to the power of c equals a to the power of b c end exponent close parentheses$
Likewise, $5 to the power of 1 fourth end exponent equals 5 to the power of fraction numerator 1 cross times 3 over denominator 4 cross times 3 end fraction end exponent equals 5 to the power of 3 over 12 end exponent$
$not stretchy rightwards double arrow open parentheses 5 cubed close parentheses to the power of 1 over 12 end exponent equals 125 to the power of 1 over 12 end exponent$ (Since ) $open open parentheses a to the power of b close parentheses to the power of c equals a to the power of b c end exponent close parentheses$
Now, we have $64 to the power of 1 over 12 end exponent comma space 9 to the power of 1 over 12 end exponent$ and $125 to the power of 1 over 12 end exponent$
So, here $125 greater than 64 greater than 9$
$not stretchy rightwards double arrow 125 to the power of 1 over 12 end exponent greater than 64 to the power of 1 over 12 end exponent greater than 9 to the power of 1 over 12 end exponent$
$not stretchy rightwards double arrow fourth root of 5$ is greater

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