Q: What is the smallest positive integer value of x for which 54x is a perfect square?

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The integer is 26

Only if the integer is a perfect square.

Irrational. The square root of a positive integer is either an integer (that is, if the integer is a perfect square), or an irrational number.

Yes. The square root of a positive integer can only be an integer (if your integer is a perfect square), or an irrational number (if it isn't).

In terms of prime factors, 1008 = 24*32*7 Then since 24 and 32 are perfect squares, all that is required is to make 7q a perfect square and so q = 7.

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6.

The integer is 26

63= 9* 7. 9 is already a perfect square, so mulitiply by 7. 7 is your answer.

the answer is 144, it is divisible by 1, 4, 9, 16, 36, and 144.

its 81

Only if the integer is a perfect square.

Irrational. The square root of a positive integer is either an integer (that is, if the integer is a perfect square), or an irrational number.

Yes. The square root of a positive integer can only be an integer (if your integer is a perfect square), or an irrational number (if it isn't).

In terms of prime factors, 1008 = 24*32*7 Then since 24 and 32 are perfect squares, all that is required is to make 7q a perfect square and so q = 7.

The prime factorisation of 248832 is 2¹⁰ × 3⁵ Every perfect square number has a prime factorisation where each prime is to an even power. 2 has an even power 3 has an odd power, so need an extra power → multiple 248832 by 3 which gets (2⁵ × 3³)²

17

The answer is 19.