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.


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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 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

FAQ (Frequently Asked Questions)

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 xn = 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.