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Square roots only have two solutions if the number is positive, if the number is negative it has no solutions.

Actually ALL numbers (including negative numbers) have two square roots and three cube roots - just not all of them are real numbers; some are complex numbers.

What is a complex number?

A complex number is a number which is the sum of two parts: a real part and an imaginary part, ie are of the form (a + bi):

a is the real part;

bi is the imaginary part which is a real number (b) multiplied by the square root of -1 which is an imaginary value. To avoid having to write √-1 all the time, the little 'i' is used instead, ie i² = -1.

Now the square roots of negative numbers can be found:

eg √-4 = √(4 × -1) = √4 √-1 = 2i → the square root of -4 is ±2i.

What about cube roots?

³√8 = 2 is the real root, but there are two further complex roots: (-1 + √3 i) and (-1 - √3 i). I'll cube the first to show it does indeed equal 8 (remember that i² = -1)

(-1 + √3 i)³ = (-1 + √3 i)(-1 + √3 i)(-1 + √3 i)

= (-1 + √3 i)((-1)×(-1) + -1 × √3 i + -1 × √3 i + (√3 i)×(√3 i))

= (-1 + √3 i)(1 - 2 (√3 i) + 3i²)

= (-1 + √3 i)(1 - 2 (√3 i) + 3 × -1)

= (-1 + √3 i)(1 - 2 (√3 i) - 3)

= (-1 + √3 i)(-2 - 2 (√3 i))

= (-1 + √3 i)(-1 - √3 i) × 2

= ((-1)² - (√3 i)²) × 2

= (1 - (-3)) × 2

= (1 + 3) × 2

= 4 × 2

= 8

Similar cubing of (-1 - √3 i) equals 8.

Q: Why do square roots have two solutions and cube roots only have one solution?

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For school you will need to learn how to find square and cube roots in order to have the needed prerequisites to answer progressively harder and more complex problems.

For school you will need to learn how to find square and cube roots in order to have the needed prerequisites to answer progressively harder and more complex problems.

there is no cube roots in negative

When (if) you learn more advanced mathematics you will find that there are, in fact 3 cube roots for any non-zero number (in the complex field). In general, there are n nth roots (de Moivre's theorem). However, only one of the cube roots can be a real number, the other two are complex numbers. The reason is that the product of a pair of negative numbers is positive. As a result both x and -x are square roots of x^2. But the product of three negative numbers is itself negative, so for cube roots the signs match up.

Simplest radical form means simplifying a radical so that there are no more square roots, cube roots, 4th roots and such left to find. It also means removing any radicals in the denominator of a fraction.

Related questions

The answer depends on "different from WHAT?" Positive cube roots, or negative square roots?

For school you will need to learn how to find square and cube roots in order to have the needed prerequisites to answer progressively harder and more complex problems.

an alg expression involving square roots, cube roots, etc

4

The ancients - Egyptians or Greeks. They probably came across the square root of 2 when considering the diagonal of a square with sides of length 1. The cube root of 3 would have arisen, similarly, with the principal diagonal of a unit cube.

I posted an answer about cube roots of complex numbers. The same info can be applied to square roots. (see related links)

Writing out the solution for the rubicks cube is very difficult. Check up the videos on youtube that will show you how.

There is no number that has more than two square roots.By definition, the "square" bit implies two.Every number has exactly:TWO square roots,THREE cube roots,FOUR quadratic roots,etc.

All numbers have cube roots (not necessarily integral cube roots) so every prime has cube roots.

Not sure what answer you are looking for, but here are 4 types of roots in math. First is a square roots, next is cube roots, then the nth roots, and lastly rational roots.

Yes.