# Nth Root

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## 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.

### 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 x0, 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 0n 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: 2xy2