## What is Nth Root?

In Mathematics, the nth root of a number x is a number y which when raised to the power n, obtains x:

yⁿ = x

Here, n is a positive integer, sometimes known as the degree of the root. A root of degree 2 is known as a square root, whereas the root of degree 3 is known as a cube root. Roots of higher degree are also referred to using ordinary numbers as in fourth root, fifth root, twentieth root, etc. The calculation of the nth root is a root extraction.

For example, 4 is a square root of 2, as 22 = 4, and −2 is also a square root of 4, as (−2)2 = 4.

\[\sqrt{x} \times \sqrt{x} = x\] Here, the square root is used twice in multiplication to get the original value.

\[\sqrt[3]{x} \times \sqrt[3]{x} = x\] Here, cube root is used thrice in multiplication to get the original value.

\[\sqrt[n]{x} \times \sqrt[n]{x} . . . \sqrt[n]{x} = x\] Here, the nth root is used n times in multiplication to get the original value.

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### Nth Root Definition

Recall that k is a square root of y if and only if k² = y. Similarly, k is a cube root of y if and only if k³ = y. For example, 5 is a cube root of 125 because 5³ = 125. Let us understand the nth root definition with this concept.

Let n be an integer greater than 1, then y is the nth root of x if and only if yⁿ = x.

For example, -1/2 is the 5th root of -1/32 as (-1/2)⁵ = -1/32. There are no special names given to the nth root other than the square root (where n = 2), and the cube root (where n = 3). Other nth roots are known as the fourth root, fifth root, and so on.

### Nth Root Symbol

The symbol used to represent the nth root is \[\sqrt[n]{x}\]. It is a radical symbol used for square root with a little n to define the nth root.

In expression \[\sqrt[n]{x}\], n is known as the index and the x is known as the radicand.

### How to Find the Nth Root of a Number?

The nth root of a number can be calculated using the Newton method. Let us understand how to find the nth root of a number, ‘A’ using the Newton method.

Start with the initial guess x_{0}, and then repeat using the recurrence relation.

\[x_{k+1} = \frac{1}{n}(n - 1)x_{k} + \frac{A}{X_{k^{n+1}}})\] until the desired precision is reached.

On the basis of the application of nth root, it may be adequate to use only the first Newton approximant: \[\sqrt[n]{x^{n} + y} \approx x + \frac{y}{nx^{n-1}}\].

For example, to find the fourth root of 16, note that 2⁴ = 16 and hence x = 2, n = 4, and y = 2 in the above formula. This yields:

\[\sqrt[5]{34} = \sqrt[5]{32 + 2} \approx 2 + \frac{2}{5.16} = 2.025\]. The error in the approximation is only about 0,03%.

### When Does the Nth Root Exist?

In a real number system,

If n is an even whole number, the nth root of x exists whenever x is positive, and for all x.

If n is an odd whole number, the nth root of x exists for all x.

Example:

\[\sqrt[4]{-81}\] is not a real number whereas,

\[\sqrt[5]{-32} = -2\]

Things get more complicated in the complex number system.

Every number has a square root, cube root, fourth root, fifth root, and so on.

Example:

The fourth root of a number 81 are 3, -3, 3i, -3i, because

3⁴ = 81

-3⁴ = 81

(3i)⁴ = 3⁴ i⁴ = 81

(-3i)⁴ = (-3)⁴ i⁴ = 81

### Properties of Nth Root

Expressing the degree of the nth root in its exponent form as in y¹ makes it easier to manipulate roots and power.

\[\sqrt[n]{a^{x}} = (a^{x})^{1/y} = a^{x/y}\]

There is exactly one positive nth root in every positive real number. Hence, the rules of operation with surds including positive radicand x, and y are straightforward within a real number.

\[\sqrt[n]{xy} = \sqrt[n]{x} \sqrt[n]{y}\]

\[\sqrt[n]{\frac{x}{y}} = \frac{\sqrt[n]{x}}{\sqrt[n]{y}}\]

Subtleties can take place while calculating the nth root of a negative or complex number. For example,

\[\sqrt{-1} \times \sqrt{-1} = \sqrt{-1 \times -1} = 1\]

But instead , \[\sqrt{-1} \times \sqrt{-1} = i \times i = i^{2} = -1\]

As the rule \[\sqrt[n]{x} \sqrt[n]{y} = \sqrt[n]{xy}\], strictly valid for non-negative real radicands only, its use leads to inequality in step 1 above.

### Facts to Remember

The nth root of 0 is 0 for all positive integers n, as 0

^{n}is equal to 0.The nth root of 1 is known as roots of unity and plays an important role in different areas of Mathematics such as number theory, the theory of equation, etc.

### Simplifying Nth Root

Let us learn to simplify the nth root through the examples below:

1. \[\sqrt[5]{-32}\]

Solution:

The value of \[\sqrt[5]{-32}\] is -2, because (-2)⁵ = -32.

2. Find \[\sqrt[6]{64x^{6} y^{12}}\]

Solution:

Step 1: \[\sqrt[6]{64x^{6} y^{12}}\] (Given)

Step 2: \[\sqrt[6]{(2)^{6} x^{6} (y^{2})^{6}}\]

Step 3: \[\sqrt[6]{(2xy^{2})^{6}}\]

Step 4: 2xy^{2}

Q1. What is the Root in Maths?

Ans. In Mathematics, a root is a solution to an equation, usually represented as an algebraic expression or formula. If k is a positive real number and n is a positive integer, then there includes a positive real number x such that x^{n} = k. Hence, the principal nth root of x is expressed as ^{n}√x. The integer n is known as the index of the root.

Q2. What is Known as the Principal nth Root of a Number?

Ans. If x is any positive integer with at least one nth root, then the principal nth root of x, represented as ^{n}√x is the number with the same sign as x, that when raised to the nth power equals x. Here, the index of radical is n.

Q3. How do the Roots of a Real Number Represent?

Ans. The roots of a real number are represented using the radical symbol or radix √, with √a representing the positive square root of ‘a’ if ‘a’ is positive for; higher roots and ^{n}√a represents the real nth root if n is odd, and positive nth root if n is even and a is positive. In other ways, the symbol is not commonly used as ambiguous. In the expression, ^{n}√a, the integer n is known as an index, and ais known as radicand.