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Why do square roots have two solutions and cube roots only have one solution?
Wiki User
∙ 8y ago
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.
Wiki User
∙ 8y ago
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Why would I need to use square roots and cube 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.
Why would you need to use square roots and cube root?
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.
How many real cube roots does any positive number have?
there is no cube roots in negative
Why are there two integer answers for square roots but only one integer answer for cubed roots?
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.
What is simplest radical form?
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.
How are positive square roots different?
The answer depends on "different from WHAT?" Positive cube roots, or negative square roots?
Why would I need to use square roots and cube 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.
What is radical expression?
an alg expression involving square roots, cube roots, etc
What are the 3 cube roots of -1?
4
Who created square roots and cube roots?
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.
Why would you need to use square roots and cube root?
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.
What do you think is true of the square roots of a complex number?
I posted an answer about cube roots of complex numbers. The same info can be applied to square roots. (see related links)
No of solutions of rubik's cube?
Writing out the solution for the rubicks cube is very difficult. Check up the videos on youtube that will show you how.
What is a number that has more than two square roots?
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.
What is prime with cube roots?
All numbers have cube roots (not necessarily integral cube roots) so every prime has cube roots.
What are the 4 kinds of 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.
Do negative numbers have cube roots cube roots?
Yes.
