Nth Root

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

\[y^{n} = x\] 

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 \[2^{2}\] = 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 \[k^{2}\] = y. Similarly, k is a cube root of y if and only if \[k^{3}\] = y. For example, 5 is a cube root of 125 because\[5^{3}\] = 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 \[\left ( \frac{-1}{2} \right )^{5}\] = -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 roots 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.

In order to understand the definition of the nth root more precisely, the student needs to be aware of a few other topics that will play a major role in the understanding of the nth root. These topics are explained briefly below. 

Real Numbers

Real numbers are referred to as the combination of rational and irrational numbers. All the arithmetic functions are said to be performed on these numbers and they can also be represented on the number line. 

While, on the other hand, imaginary numbers are those that cannot be expressed on a number line, and are usually used to represent complex numbers. Real numbers can be both positive or negative and are usually denoted using the letter R. the natural numbers, fractions, and decimals fall under this category. 

Rational Numbers

Rational numbers fall under the heading of real numbers. These are represented as p/q, where q is not equal to 0. Any fraction that is a non-zero denominator is termed a rational number. For example,\[\frac{1}{3}\], 1/5,\[\frac{3}{4}\], etc. in fact the number 0 can also be called a rational number as it can be written in various forms like 0/1, 0/2, 0/3, etc. but it is to be kept in mind that 1/0, 2/0, 3/0, etc are not rational as they provide us with infinite values. 

Irrational Numbers

Irrational numbers refers to the real numbers that cannot be expressed in the form of a fraction. It cannot be denoted in the form of a ratio p/q, where the letters p and q refer to integers and q is not equal to zero. One can say that it is the opposite of the rational numbers. 

Irrational numbers are normally expressed in the form R∖Q. The backward slash refers to the ‘set minus’. It is also often expressed in the form of R-Q, which refers to the difference between a set of real numbers and a set of irrational numbers. 

Complex Numbers

Complex numbers are referred to as numbers that can be expressed in the form of a + ib. a, b are the real numbers, while i refers to the imaginary numbers. For instance, 2+3i is a complex number where 2 is a real number while 3i denotes the imaginary number.

The imaginary number is always denoted with the alphabet i or j which is equal to \[\sqrt{-1}\], where \[i^{2} = -1\].

Square Roots

A square root of the number r can be referred to as x, which when squared, gives the result r. 

\[r^{2} = x\]

It is to be noted that every positive real number possesses two square roots, one that is positive and one that is negative. For instance, the number 25 has two square roots, one is 5 and the other is -5. The positive square root is also denoted as the principal square root.

As the square root of every number is non negative, the negative numbers do not possess a square root. But every negative real number has two imaginary square roots associated with them. For example, the square root of -25 will be 5i and -5i. The i here represents the number whose square is supposed to be -1. 

Cube Roots

The cube root of a given number x can be a number r whose cube will be x.

r3=x

How to find the Nth Root of a Number?

Ans: 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 \[2^{4}\] = 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

Ans: 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\[^{2}\]