3. why do cube roots of negative numbers exist in the set of real numbers,
but square roots of negative numbers do not?
a. the square root of a number, positive or negative, is always negative.
however, the cube of a negative number does not exist.
b. the square root of a number, positive or negative, is always positive.
however, the cube of a negative number is positive.
c. the square root of a number, positive or negative, is always positive.
however, the cube of a negative number is negative.
d. the square root of a number, positive or negative, is always positive.
however, the cube of a negative number does not exist.

  C. The square root of a number, positive or negative, is always positive. However, the cube of a negative number is negative.

Step-by-step explanation:

Of the choices offered, the one shown above is the only one that remotely makes any sense. Its description of square roots is erroneous, however.

The square root of a number n is the solution to the equation ...

  x² = n

The function f(x) = x² has a range that only includes non-negative numbers, so the square root of a negative number does not exist. (Your answer choice says it is positive. It is actually non-existent among real numbers.)

On the other hand, the cube root of a number n is the solution to the equation ...

  x³ = n

Since f(x) = x³ has a range that includes all real numbers, the cube root of n exists for all real numbers. (It is negative for negative values of n.)

Comment on answer wording

The answer wording would be correct if the word "root" were left out.

I'd go with

C. The square root of a number, positive or negative, is always positive.

However, the cube of a negative number is negative.

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That's not totally true and it's a bit off the point.  The square root of a negative number isn't a positive number, it's a positive number times i.

Even that's not totally true. We need to distinguish between a square root and the principal square root function.  What's true is every non zero complex number has two square roots, negations of each other. , the radical sign applied to a real number, is a function, because we define it to be by choosing one of the square roots as the principal square root.  

The right answer would be more like

E. The square of a non-zero real number is always a positive real, so considering the inverse, negative reals don't have real square roots.  The cube of a  real number has the same sign as the original number, so again considering the inverse, negative reals do have real cube roots.

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Step-by-step explanation: