x is the nth root of a number y is equivalent to xn = y.
For n = 2, the nth root is called the square root.
For n = 3, the nth root is called the cubic root.
1) Since 32 = 9, 3 is the square (n=2) root of 9.
2) Also since (-3)2 = 9, -3 is also a square root of 9.
3) Since (-2)3 = -8, -2 is the cubic (n = 3) root of -8
4) Since 34 = 81 and (-3)4 =81, the fourth roots of 81 are 3 and - 3
Properties of Roots of Real Numbers
1) For n even and y positive, there are two nth roots of y
Since 104=10000 and (-10)4 = 10000, the fourth roots of 10000 are 10 and -10.
2) For n even and y negative, there are no real nth roots of y.
The square root of - 4 is not a real number since no real number x exists such that x2 = -4
The fourth root of - 16 is not a real number since no real number x exists such that x4 = -16
3) For n odd, there is always one nth root of y.
The cubic (n=3) root of 8 is equal to 2.
The fifth root (n=5) of -100000 is equal to -10
For n even, the principal root is the positive root. For n odd there is only one root and it is the principal root.
The principal 6th of 64 is equal to 2.
The principal cubic root of -64 is equal to - 4.
The symbol √ is called a radical and is used to indicate the principal root of a number as follows:
where n is called the index of the radical and y is called the radicand.
Because of its widespread use, the square root of y is written as √y without indicating the index.
Questions With Answers
Use a calculator to approximate the following to 3 decimal places: