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Maths-

General

Easy

Question

Arrange in ascending order of magnitude $square root of 5 comma root index 8 of 11$ and $2 root index 6 of 3$

hint Hint:

Use LCM property.

The correct answer is: ascending order

Complete step by step solution:
Here we can write, $square root of 5 equals 5 to the power of 1 half end exponent comma root index 8 of 11 equals 11 to the power of 1 third end exponent text and end text 2 root index 6 of 3 equals root index 6 of 12 equals 12 to the power of 1 over 6 end exponent$
On taking the LCM of 2,3,6 we have LCM as 6.
So, $not stretchy rightwards double arrow 5 to the power of 1 half end exponent equals 5 to the power of fraction numerator 1 cross times 3 over denominator 2 cross times 3 end fraction end exponent equals 5 to the power of 3 over 6 end exponent$
$not stretchy rightwards double arrow open parentheses 5 cubed close parentheses to the power of 1 over 6 end exponent equals 125 to the power of 1 over 6 end exponent space of 1em open parentheses text Since end text open parentheses a to the power of b close parentheses to the power of c equals a to the power of b c end exponent close parentheses$
Likewise, $not stretchy rightwards double arrow 11 to the power of 1 third end exponent equals 11 to the power of fraction numerator 1 cross times 2 over denominator 3 cross times 2 end fraction end exponent equals 11 to the power of 2 over 6 end exponent$
$not stretchy rightwards double arrow open parentheses 11 squared close parentheses to the power of 1 over 6 end exponent equals 121 to the power of 1 over 6 end exponent space of 1em open parentheses text Since end text open parentheses a to the power of b close parentheses to the power of c equals a to the power of b c end exponent close parentheses$
Likewise, $not stretchy rightwards double arrow 12 to the power of 1 over 6 end exponent$
Now, we have $125 to the power of 1 over 6 end exponent comma 121 to the power of 1 over 6 end exponent$ and $12 to the power of 1 over 6 end exponent$
So, here ascending order is 12 < 121 < 125
$not stretchy rightwards double arrow 12 to the power of 1 over 6 end exponent less than 121 to the power of 1 over 6 end exponent less than 125 to the power of 1 over 6 end exponent$
$table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell not stretchy rightwards double arrow 12 to the power of 1 over 6 end exponent less than 11 to the power of 1 third end exponent less than 5 to the power of 1 half end exponent end cell row cell not stretchy rightwards double arrow root index 6 of 12 less than cube root of 11 less than square root of 5 end cell end table$

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