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Find a number $x$ such that $x 2$ is an irrational number but $x 4$ is a rational number.

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3

33

3

34

answer is A.

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Detailed Solution

To find such a number, we take a number whose fourth power makes the number rational but the second power does not.
From option (1),
Let’s assume

x = 34 .

We have,

x 4 = (34) 4 = 3,

which is a rational number.
Also,

x 2 = (34) 2 = 3,

which is irrational.
Therefore, the value of

x

could be

34 .

From option (2),
Let’s assume

x = 3 .

We have,

x 4 = 34 = 9,

which is a rational number.
Also,

x 2 = 32 = 3,

which is also a rational number.
Therefore, the value of

x

couldn’t be

3 .

From option (3),
Let’s assume

x = 33 .

We have,

x 4 = (33) 4 = 333,

which is an irrational number.
Also,

x 2 = (33) 2 = 323,

which is also irrational.
Therefore, the value of

x

couldn’t be

33 .

From option (1),
Let’s assume

x = 3.

We have,

x 4 = (3) 4 = 81,

which is a rational number.
Also,

x 2 = (3) 2 = 9,

which is also rational.
Therefore, the value of

x

couldn’t be 3.
Hence, option (1) is correct.

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Find a number x such that x 2 is an irrational number but x 4 is a rational number.

Detailed Solution

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Find a number $x$ such that $x 2$ is an irrational number but $x 4$ is a rational number.