Question 99554
{{{sqrt(77)}}} is irrational. For a number to be rational, it must be able to be expressed as a fraction of two integers {{{a/b}}} in lowest terms(prime numbers).
One major thing to know, and you should investigate this for yourself, is that there is no prime number that will have a rational square root. 77 is a product of 11 and 7, both are primes and that means that it will not have a rational square root.
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Is {{{sqrt(-x)}}} rational or irrational? Well, that all depends on the x. It will definitely have a complex root(imaginary), but that doesn't mean that it will be irrational. It all depends on the x for the same reason in the first question. 
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Is that a right triangle? And what do you mean by altitude in regards to the triangle?
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The rule is {{{sqrt(a/b) = sqrt(a)/sqrt(b)}}} so your answer is {{{sqrt(9)/sqrt(16) = 3/4}}}. This also applies to your next question.

The cube root of 1/64. Same thing as above except for cube roots. The cube root of 1 = 1; the cube root of 64 = 4. So the cube root of 1/64 = 1/4.