Square Root, Cubic Root, N-th Root Assignment Help
Introduction to Square Root, Cubic Root and nth Root:
Square root of a given number is another, non-negative, number that, when squared, equal to the given number. It is written as √x, x=(√x)2. A root of degree 2 is called a square root. Roots of higher degree are referred by using ordinal numbers, as in fourth root, twentieth root, etc.
Cubic root: The cube root of a real number x, written as x√3. A root of degree 3, is called cubic root. For example, 3√27=3. The cube root notation is actually an alternative to exponentiation. That is, x√3=x1/2.
Properties of the cube root:
- The cube root operation is distributive for multiplication and division, but not for addition and subtraction.
- The cube root is a special case of the general nth root.
- The cube root is a continuous mapping from RR.
- The cube root function from RR defined as f(x)=x3 is an odd function.
Nth root: A root of n-th degree of a given number is another number that, when taken to n-th degree, is equal to a given number: x=(n√x)n
The nth root of a number x is a number r which, when raised to the power of n, equals x
where n is the degree of the root.
2 is a square root of 4, since 22 = 4.
2 is also a square root of 4, since (2)2 = 4.
Identities and properties:
Every positive real number has a positive nth root and the rules for operations with such surds are straightforward:
nab = na nb,
na/b = na/nb.
Using the exponent form as in x1/n normally makes it easier to cancel out powers and roots.
nam = (am)1/n = am/n.
Problems can occur when taking the nth roots of negative or complex numbers. For instance:
-1 * -1 = -1
-1*-1 = 1
when taking the principal value of the roots. See failure of power and logarithm identities in the exponentiation article for more details.
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