Question

How would you simplify a negative square root?

Provide a detailed explanation, with an example, to receive full credit.

Answer 1

Explanation:

The square root of a number A, is a number B such that, when it is multiplied by itself, the result is A.

If A × A = B

Then √B = A.

Now the multiplication of two numbers gives a positive number if both numbers are positive, or both numbers are negative.

2 × 2 = -2 × -2 = 4

3 × 3 = -3 × - 3 = 9

And so on.

So, the square root of 4 = 2 or -2

The square root of 9 = 3 or -3

But if one of the numbers is positive while the other is negative, then the result is negative.

2 × -2 = -4

3 × -3 = -9

Clearly, √(-4) ≠ 2 ≠ -2

√(-9) ≠ 3 ≠ -3

It is impossible to find the square root of negative numbers on the real line. This gives rise to the introduction of Complex Number.

Let i² = -1, then we have that

√(-1) = i.

This is the idea of Complex number, and it helps solve the problem of the negative square roots, and every negative number can be written as the multiplication of -1 and the inverse of the number.

-A = -1 × A

So, √(-A) = √(-1 × A)

= √(-1) × √A

= i × √A

= i√A

Example, to simplify √(-16)

√(-16) = √(-1 × 16)

= √(-1) × √16

= i × ±4

= ±4i

Answer 2

To simplify

√(-x) = √((x)(-1)) = √((x)(i^2))

√(-x) = √(i^2) × √x = i√x

For example;

Simplify √-9

√-9 = √(-1×9) = √-1 × √9

√-9 = √(i^2) × √9 = i × 3

√-9 = 3i

Explanation:

Given a negative square root √(-x);

From our knowledge of complex numbers, we know that

i^2 = -1 and vise versa

To simplify

√(-x) = √((x)(-1)) = √((x)(i^2))

√(-x) = √(i^2) × √x = i√x

For example;

Simplify √-9

√-9 = √(-1×9) = √-1 × √9

√-9 = √(i^2) × √9 = i × 3

√-9 = 3i