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Can a radical be negative when the index is even?

  • Last Updated : 07 Dec, 2021

Complex numbers are the combination of real values and imaginary values. They are expressed in the form of x + iy where x and y are real numbers and i is the imaginary part also called iota. It is often represented by z. The value ‘x’ is called the real part, denoted by Re(z) and the value ‘y’ is called the imaginary part which is denoted by Im(z). In complex numbers, one part is purely real and the other part is purely imaginary. 

Real and Imaginary numbers

Real Numbers are those numbers whose square gives a positive result. They can be positive, negative, integers, rational, irrational, etc. They can be represented on a number line. It is represented by Re().

Imaginary Numbers are those numbers whose square gives a negative value. They cannot be represented on a number line. They are denoted by Im(). The imaginary numbers are of the form ‘bi’ where i is the iota and b is the real number. Example: z = 1 + 4i. Here in the above example, it is of the form a + ib where a = 1 and b = 4 which are real numbers.

  • Re(z) = 1
  • Im(z) = 4

More About Iota

An imaginary number is denoted by iota ‘i’. The ‘i’ used in complex numbers is known as iota. It is used to find the square root of negative numbers. Value of i = √(-1). If the square operation of i is performed,

  1. i2 = i.i = -1
  2. i3 = i.i.i = -i
  3. i4 = 1

Radicals and Index

Radicals mean root. It is often known by the name radix. Any expression that is expressed under the radical sign ( √ ) is known as radical expression. A radical expression can contain any algebraic or numerical expression. An index is a number that helps to calculate the nth root. Here n is the index. The index and radical are expressed in the form:

index√ radical

Can radical be negative when the index is even?

Answer:

Yes, radical can be negative when the index is even. But it leads to a new theory which is known as complex numbers. In this the root becomes imaginary. It cannot be represented on the number line. 

For example, to find the square root of -9, the index is 2 which is even and radical is -9 which is negative. The result will be 3i or -3i which is an imaginary number.

Another example is to find the sixth root of -729. Here the radical is negative and the index is 6 which is even. The result will be 3i which is a complex number.

Similar Problems 

Question 1: Find the fourth root of -16.

Solution: 

Here the index is 4 and radical is -16

As known 24 = 16

Since the roots can be positive or negative if the index is of the order 2n. So the result is 2i or -2i

Question 2: Find the square of 6i.

Solution: 

As known i.i= -1

So, 6i × 6i = -36

Question 3: Find the square root of -4.

Solution: 

As known the square root of 4 is 2.

Since the roots can be positive or negative if the index is of the order 2n. So, the result is 2i or -2i.

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