Q: What is the smallest perfect cube and perfect square to the eleventh power?

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Divide 2880 by 5 to give the perfect square 576 = 242. First write 2880 in its prime factorisation in power form: 2880 = 26 x 32 x 51 To be a perfect square, all the prime factors need to have an even power, so to find the smallest number by which to divide to get a perfect square only all the odd powers need to be reduced by 1; only the power of the prime factor 5 is odd and needs to be reduced by 1, so divide by 5.

0, 1To be both a square and a cube, the number has to be a sixth power. You don't need to calculate though, since 06=0 and 16=1.The next 2 sixth powers are 26=64 and 36=729.

To be a perfect square, all the primes in a number's prime factorisation must have an even power To be a perfect cube, all the primes in a number's prime factorisation must a power that is a multiple of 3 → To be a perfect square, all the primes in a number's prime factorisation must a power that is a multiple of 3 and a multiple of 2, ie the power must be a multiple of 6 The smallest prime is 2 2⁶ = 64 = (2³)² = 8² = (2²)³ = 4³ 2¹² = 4096 (too large) 3⁶ = 729 (too large) There is also 1 = 1² = 1³ Thus the whole numbers less than 100 which are both perfect squares and perfect cubes are 1 and 64.

Sure. Take the fourth power of any natural number, for example 24 = 16. It square root is a perfect square, in this case 22. As another example, the square root of 34 = 32.

It is the perfect square of that number.

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the answer is definitly 0 (zero)!

Divide 2880 by 5 to give the perfect square 576 = 242. First write 2880 in its prime factorisation in power form: 2880 = 26 x 32 x 51 To be a perfect square, all the prime factors need to have an even power, so to find the smallest number by which to divide to get a perfect square only all the odd powers need to be reduced by 1; only the power of the prime factor 5 is odd and needs to be reduced by 1, so divide by 5.

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

8

In the prime factorisation of a perfect square in power format, all primes must be to an even power. 5688 = 23 x 32 x 79 To convert this it a perfect square, it needs the 2 and 79 to have an even power - the 3 is already at an even power. The smallest way to do this is to multiply 5688 by 2 x 79, ie multiply it by 158. 5688 x 158 = 23 x 32 x 79 x 2 x 79 = 24 x 32 x 792 = (22 x 3 x 79)2 = (948)2 = 898704

81

Three to the seventh power

0, 1To be both a square and a cube, the number has to be a sixth power. You don't need to calculate though, since 06=0 and 16=1.The next 2 sixth powers are 26=64 and 36=729.

To be a perfect square, all the primes in a number's prime factorisation must have an even power To be a perfect cube, all the primes in a number's prime factorisation must a power that is a multiple of 3 → To be a perfect square, all the primes in a number's prime factorisation must a power that is a multiple of 3 and a multiple of 2, ie the power must be a multiple of 6 The smallest prime is 2 2⁶ = 64 = (2³)² = 8² = (2²)³ = 4³ 2¹² = 4096 (too large) 3⁶ = 729 (too large) There is also 1 = 1² = 1³ Thus the whole numbers less than 100 which are both perfect squares and perfect cubes are 1 and 64.

Sure. Take the fourth power of any natural number, for example 24 = 16. It square root is a perfect square, in this case 22. As another example, the square root of 34 = 32.

16

1