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=x^1/2.

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 R→R.
• The cube root function from R→R defined as f(x)=x√3 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

rⁿ=x,

where n is the degree of the root.

For example:

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:

n√ab = n√a n√b,

n√a/b = n√a/n√b.

Using the exponent form as in x^1/n normally makes it easier to cancel out powers and roots.

n√a^m = (a^m)^1/n = a^m/n.

Problems can occur when taking the nth roots of negative or complex numbers. For instance:

√-1 * √-1 = -1

whereas

√-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.