Just as numbers can be square and cubic, roots can also be square and cubic.
The symbol |\sqrt{\phantom{2}}| is called a radical, or root. Its name can change depending on the number associated with it.
-
|\sqrt{x}| or |\sqrt[2]{x}| is the square root of the number |x.|
-
|\sqrt[3]{x}| is the cube root of the number |x.|
-
|\sqrt[4]{x}| is the fourth root of the number |x.|
-
|\sqrt[n]{x}| is the nth root of the number |x.|
The number or algebraic expression inside the radical is called the radicand.
Given |\{x,y\} \subseteq \mathbb{R}|, the square root of a number |y| corresponds to the positive real number |x| that, when squared, results in |y| .
||\text{If} \ x \geq 0 \ \text{and} \ x^2=y, \ \text{then} \ \sqrt{y} = x||
It is a mistake to assume that the solution of a square root |\sqrt{a}| can be negative. In fact, the question is whether the coefficient in front of the root is positive or negative.
Consider, |9 = 3^2 \ \text{and} \ 9 = (-3)^2|.
Thus,
|| +\sqrt{9} = 3 \ \text{and} \ -\sqrt{9} = -3 ||
Looking at the square root function, the negative result is obtained due to the value of the parameter |a| which is negative. By convention, the quantity |\sqrt{a}| is always positive, but we must always consider the two possible roots |\pm \sqrt{a}|.
Therefore, the notion of square root and squared numbers are closely related. In fact, taking the square root is the reverse operation of squaring a number. This relationship is useful for finding a missing value in algebra.
However, not all real numbers have a real square root.
When performing calculations only in |\mathbb{R}|, it is impossible to calculate the square root of a negative number.
||\sqrt{-25} \not\in \mathbb{R}||
However, it is possible to find a numerical solution to such a root, but the solution will be part of the set of complex numbers |(\mathbb{C}).|
Given |\{x,y\} \subseteq \mathbb{R}|, the cube root of a number |y| corresponds to a real number |x| that, when cubed, results in |y|.
||\text{If} \ (x)^3=y, \ \text{then} \ \sqrt[3]{y} = x||
Unlike the square root of a number, it is possible to calculate the cube root of any number in the set of real numbers. In addition, the real solution to a cube root is always unique.
||\text{If}\ (\text{-}3)^3 = \text{-}27, \ \text{then} \ \sqrt[3]{\text{-}27} = \text{-}3||
Based on the definition, taking the cube root of a number is the inverse operation of cubing a number. Furthermore, it is possible to use this relationship to find missing measurements in algebra.